CLASS 8 NCERT WORKSHEET CH-4 QUADRILATERALS WITH ANSWER KEY

 CLASS 8 NCERT WORKSHEET  CH-4 QUADRILATERALS WITH ANSWER KEY

Here is a worksheet based on the provided source PDF, "ch4 quadrilaterals class 8.pdf". This worksheet covers key definitions, properties, deductions, and problem-solving elements from the chapter.

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Quadrilaterals: Chapter 4 Worksheet

Instructions: Read the questions carefully and answer them based on the information provided in your textbook. Show your reasoning where required.

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Section 1: Introduction to Quadrilaterals

1. What is the derivation of the word 'quadrilateral'? Provide the meaning of its root words.
2. Observe Figures (i), (ii), and (iii) in the source. What makes these figures quadrilaterals, as opposed to Figures (iv) and (v)?

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Section 2: Rectangles and Squares

1. Define a rectangle based on its angles and opposite sides.
2. Describe two key properties of the diagonals of a rectangle, as derived through geometric deduction.
3. Explain how a carpenter can use the properties of diagonals to construct a rectangular frame, given one 8 cm long strip of wood.
   • What should be the length of the other strip?
   • Where should they both be joined?
4. A quadrilateral has diagonals that are equal in length and bisect each other.
   • If the angle between the diagonals is 60°, what type of quadrilateral is formed? Justify your answer by calculating all its interior angles.
   • Will the quadrilateral remain a rectangle if this angle is changed? Explain your reasoning using the general case with angle 'x' between the diagonals.
5. State the simplified definition of a rectangle based solely on its angles. Prove why a quadrilateral with all angles equal to 90° must have opposite sides of equal length.
6. List four properties of a rectangle.
7. Define a square.
8. Explain the relationship between squares and rectangles using a Venn diagram.
9. What additional condition, beyond equal and bisecting diagonals, is necessary for a quadrilateral to be a square (instead of just a rectangle)?
10. If the diagonals of a quadrilateral are equal in length, bisect each other, and intersect at right angles (90°), what type of quadrilateral is formed? Justify your answer.
11. List five properties of a square.
12. Problem Solving (from "Figure it Out"):
    • Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of 90°. What shape do you get?
    • If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Justify your answer using geometric reasoning.

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Section 3: Angles in a Quadrilateral

1. What is the sum of all interior angles in any quadrilateral? Explain how this can be deduced by dividing a quadrilateral into triangles.
2. Based on your answer to Question 1, explain why it is impossible for a quadrilateral to have three right angles with the fourth angle not being a right angle.

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Section 4: More Quadrilaterals with Parallel Opposite Sides (Parallelograms)

1. Define a parallelogram.
2. Is a rectangle a parallelogram? Justify your answer.
3. Illustrate the relationship between rectangles, squares, and parallelograms using a Venn diagram.
4. If one angle of a parallelogram is 30°, determine the measures of the remaining three angles. Explain the properties of adjacent and opposite angles in a parallelogram.
5. List four properties of a parallelogram.
6. Are the diagonals of a parallelogram always equal in length?
7. Prove that the diagonals of a parallelogram always bisect each other.
8. True/False: A quadrilateral whose diagonals bisect each other must be a parallelogram. Justify your answer.

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Section 5: Quadrilaterals with Equal Sidelengths (Rhombus)

1. Define a rhombus.
2. Explain why every rhombus is also a parallelogram.
3. List six properties of a rhombus.
4. Do the diagonals of a rhombus always intersect at a 90° angle? Prove your answer using congruence.
5. True/False: A quadrilateral whose diagonals are perpendicular to each other must be a rhombus. Justify your answer.
6. Problem Solving (from "Figure it Out"): Using diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.

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Section 6: Kite and Trapezium

1. Define a kite and list its key properties related to sides and diagonals.
2. Define a trapezium.
3. In a trapezium, if one pair of opposite sides are parallel, what can you say about the sum of the angles on the same side of a non-parallel transversal?
4. What is an isosceles trapezium? What special property do its base angles possess?
5. True/False: Isosceles trapeziums are parallelograms. Justify your answer.
6. Venn Diagram Challenge: Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares, illustrating their relationships. Based on your diagram and source information, answer the following:
   • What is the quadrilateral that is both a kite and a parallelogram?
   • Can there be a quadrilateral that is both a kite and a rectangle?
   • Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?

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Section 7: General Properties and Review

1. Consider a quadrilateral where the opposite sides are equal. What type of quadrilateral is it? Justify your answer.
2. True/False Statements: For each statement, state whether it is True or False and provide a justification based on the source material:
   • A quadrilateral whose diagonals are equal and bisect each other must be a square.
   • A quadrilateral having three right angles must be a rectangle.
   • A quadrilateral in which the opposite angles are equal must be a parallelogram.

A quadrilateral in which all the angles are equal is a rectangle.

Quadrilaterals: Chapter 4 Worksheet - Answer Key

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Section 1: Introduction to Quadrilaterals

The word ‘quadrilateral’ is derived from Latin words: ‘quadri’ meaning four, and ‘latus’ referring to sides.
Figures (i), (ii), and (iii) are quadrilaterals because their angles are defined as the angles between their sides, as marked in the figures. Figures (iv) and (v) do not fit this description of a quadrilateral.

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• Section 2: Rectangles and Squares

A rectangle is defined as a quadrilateral in which:

The angles are all right angles (90°).
The opposite sides are of equal length.
Alternatively, a rectangle can be defined as a quadrilateral whose diagonals are equal and bisect each other.

Two key properties of the diagonals of a rectangle are:

They always have the same length.
They always intersect at their midpoints, meaning they bisect each other.

To construct a rectangular frame, the carpenter should:

Make the length of the other strip also 8 cm long, as the diagonals of a rectangle must have the same length.
They should both be joined at their midpoints, as the diagonals of a rectangle bisect each other.

A quadrilateral has diagonals that are equal in length and bisect each other:

If the angle between the diagonals is 60°:In ∆AOB, where OA=OB and the angle at O is 60°, the base angles 'a' are calculated as (180° - 60°)/2 = 60°.
Similarly, for ∆AOD, if the angle is 120° (linear pair to 60°), the base angles 'b' are (180° - 120°)/2 = 30°.
The interior angles of the quadrilateral are formed by a+b, which is 60° + 30° = 90°.
Since all angles are 90° and opposite sides are equal (due to congruent triangles formed by the diagonals), the quadrilateral formed is a rectangle.
Yes, the quadrilateral will remain a rectangle if this angle is changed. If the angle between the diagonals is 'x', the base angles of the isosceles triangles formed by the diagonals are a = 90° - x/2 and b = x/2. The sum of these angles, which forms each interior angle of the quadrilateral, is a + b = (90° - x/2) + x/2 = 90°. Therefore, no matter what the angles between the diagonals are, if the diagonals are equal and they bisect each other, the quadrilateral formed is a rectangle.

The simplified definition of a rectangle based solely on its angles is: A rectangle is a quadrilateral in which all the angles are 90°. To prove why a quadrilateral with all angles equal to 90° must have opposite sides of equal length:

Consider a quadrilateral ABCD with all angles measuring 90°.
Draw diagonal BD. In ∆BAD and ∆DCB, ∠BAD = ∠DCB = 90°.
By geometric reasoning (similar to Deduction 2), ∠1 (∠ADB) = ∠2 (∠CBD).
Therefore, ∆BAD ≅ ∆DCB by the AAS congruence condition.
As corresponding parts of congruent triangles, AD = CB and DC = BA, proving that the opposite sides have equal lengths.

Four properties of a rectangle are:

All the angles of a rectangle are 90°.
The opposite sides of a rectangle are equal.
The opposite sides of a rectangle are parallel to each other.
The diagonals of a rectangle are of equal length and they bisect each other.
A square is a quadrilateral in which all the angles are equal to 90°, and all the sides are of equal length.
Every square is also a rectangle, but every rectangle is not a square. This relationship can be represented by a Venn diagram where the set of squares is entirely contained within the set of rectangles.
Beyond equal and bisecting diagonals, the additional condition necessary for a quadrilateral to be a square is that the diagonals must intersect at right angles (90°).
If the diagonals of a quadrilateral are equal in length, bisect each other, and intersect at right angles (90°), the type of quadrilateral formed is a square. This is because equal and bisecting diagonals make it a rectangle, and the additional condition of perpendicular intersection ensures that all sides are equal, thus satisfying the definition of a square.

Five properties of a square are:

All the sides of a square are equal to each other.
The opposite sides of a square are parallel to each other.
The angles of a square are all 90°.
The diagonals of a square are of equal length and they bisect each other at 90°.
The diagonals of a square bisect the angles of the square (divide the angles into equal halves).

Problem Solving (from "Figure it Out"):

If a quadrilateral's diagonals have equal lengths of 8 cm, bisect each other, and intersect at an angle of 90°, the shape you get is a square.
Yes, if a quadrilateral has four equal sides and one angle of 90°, it will be a square.
Justification: A quadrilateral with four equal sides is a rhombus. In a rhombus, adjacent angles sum to 180° and opposite angles are equal. If one angle is 90°, its adjacent angles must also be 90° (180° - 90° = 90°), and its opposite angle must be 90°. Thus, all four angles are 90°. A quadrilateral with all sides equal and all angles 90° is a square.

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• Section 3: Angles in a Quadrilateral

The sum of all interior angles in any quadrilateral is 360°. This can be deduced by:

Drawing a diagonal (e.g., SM in quadrilateral SOME) which divides the quadrilateral into two triangles (∆SEM and ∆SOM).
The sum of angles in each triangle is 180°.
Adding the angles of both triangles: (∠1 + ∠2 + ∠3) + (∠4 + ∠5 + ∠6) = 180° + 180° = 360°.
Since these six angles combine to form the four angles of the quadrilateral, the total sum is 360°.
It is impossible for a quadrilateral to have three right angles with the fourth angle not being a right angle because the sum of all angles in any quadrilateral must be 360°. If three angles are 90° each, their sum is 270°. For the total sum to be 360°, the fourth angle must be 360° - 270° = 90°.

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• Section 4: More Quadrilaterals with Parallel Opposite Sides (Parallelograms)

A parallelogram is a quadrilateral in which opposite sides are parallel.
Yes, a rectangle is a parallelogram. A rectangle has opposite sides parallel, which satisfies the definition of a parallelogram. Specifically, a rectangle is a special type of parallelogram where all its angles are 90°.
The Venn diagram shows Rectangles and Squares both nested inside the larger set of Parallelograms. The Square set is also nested inside the Rectangle set.

If one angle of a parallelogram is 30° (e.g., ∠A = 30°):

Adjacent angles in a parallelogram add up to 180°. So, ∠D = 180° - 30° = 150° and ∠B = 180° - 30° = 150°.
Opposite angles in a parallelogram are equal. So, ∠C = ∠A = 30°.
The remaining three angles are 150°, 30°, and 150°.

Four properties of a parallelogram are:

The opposite sides of a parallelogram are equal.
The opposite sides of a parallelogram are parallel.
In a parallelogram, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a parallelogram bisect each other.
No, the diagonals of a parallelogram need not be equal in length.

To prove that the diagonals of a parallelogram always bisect each other:

Consider a parallelogram EASY with diagonals AY and ES intersecting at O.
AE = YS because they are opposite sides of a parallelogram.
Angles ∠EAO and ∠YSO are equal as alternate angles (since AE||YS and AY is a transversal).
Angles ∠AEO and ∠SYO are equal as alternate angles (since AE||YS and ES is a transversal).
Therefore, ∆AOE ≅ ∆YOS by the ASA congruence condition.
As corresponding parts of congruent triangles, OA = OY and OE = OS. This means O is the midpoint of both diagonals, thus the diagonals bisect each other.
True. A quadrilateral whose diagonals bisect each other must be a parallelogram [58(iii)]. If the diagonals bisect each other, it can be proven through congruence (e.g., SAS for triangles formed by diagonals) that opposite sides are equal, which in turn implies they are parallel, making the quadrilateral a parallelogram.

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• Section 5: Quadrilaterals with Equal Sidelengths (Rhombus)

A rhombus is a quadrilateral in which all the sides have the same length.
Every rhombus is also a parallelogram because its opposite sides are parallel. This can be shown by drawing a diagonal (e.g., AE in rhombus GAME). Since the alternate interior angles formed by the diagonal and the sides are equal (due to all sides being equal), it implies that opposite sides are parallel (EM||GA and GE||AM).

Six properties of a rhombus are:

All the sides of a rhombus are equal to each other.
The opposite sides of a rhombus are parallel to each other.
In a rhombus, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a rhombus bisect each other.
The diagonals of a rhombus bisect its angles.
The diagonals of a rhombus intersect each other at an angle of 90°.

Yes, the diagonals of a rhombus always intersect at a 90° angle. Proof using congruence:

Consider the rhombus GAME with diagonals intersecting at O.
In ∆GEO and ∆MEO:GE = ME (all sides of a rhombus are equal).
EO is common to both triangles.
GO = MO (diagonals of a rhombus bisect each other).
Therefore, ∆GEO ≅ ∆MEO by the SSS congruence condition.
Since they are congruent, their corresponding angles are equal: ∠GOE = ∠MOE.
These two angles form a linear pair, meaning ∠GOE + ∠MOE = 180°.
Thus, each angle must be 90°, so the diagonals intersect at right angles.
False. A quadrilateral whose diagonals are perpendicular to each other is not necessarily a rhombus [58(iv)]. A kite also has perpendicular diagonals, but a general kite does not have all four sides equal like a rhombus.
Problem Solving (from "Figure it Out"): To construct a rhombus whose diagonals are of lengths 4 cm and 5 cm, one would use the properties that diagonals bisect each other and intersect at 90°. Draw a 5 cm line segment and mark its midpoint. Then, draw a 4 cm line segment perpendicular to the first one at its midpoint, ensuring it is bisected (2 cm on each side of the intersection). Connect the endpoints of these two perpendicular, bisecting diagonals to form the rhombus.

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• Section 6: Kite and Trapezium

A kite is a quadrilateral that can be labelled ABCD such that AB = BC (one pair of adjacent sides are equal) and CD = DA (another pair of adjacent sides are equal). Key properties of its diagonals:

The diagonal BD (connecting the vertices between the equal sides) bisects ∠ABC and ∠ADC.
The diagonal BD bisects the other diagonal AC (AO = OC), and is perpendicular to it.
A trapezium is a quadrilateral with at least one pair of parallel opposite sides.
In a trapezium, if one pair of opposite sides are parallel (e.g., PQ||SR), then the sum of the angles on the same side of a non-parallel transversal is 180°. Therefore, ∠S + ∠P = 180° and ∠R + ∠Q = 180°.
An isosceles trapezium is a trapezium where the non-parallel sides have the same lengths. Its special property is that the angles opposite to the equal sides (base angles) are equal (e.g., ∠U = ∠V in trapezium UVWX).
False. Isosceles trapeziums are not parallelograms [59(vii)]. A parallelogram requires both pairs of opposite sides to be parallel, whereas a trapezium (even isosceles) only has at least one pair of parallel opposite sides.

Venn Diagram Challenge:

The quadrilateral that is both a kite and a parallelogram is a rhombus. A rhombus has all sides equal, which satisfies the conditions of a kite (all adjacent pairs are equal) and a parallelogram (opposite sides are parallel).
Yes, there can be a quadrilateral that is both a kite and a rectangle. This quadrilateral is a square. A square is a rectangle (all angles 90°) and a kite (all adjacent sides are equal, since all sides are equal). Such a figure would have diagonals that are equal, bisect each other, and are perpendicular, which defines a square.
No, every kite is not a rhombus [55(iii)]. The correct relationship is that a rhombus is a special type of kite. While a rhombus has all four sides equal, a general kite only requires two distinct pairs of adjacent sides to be equal.

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• Section 7: General Properties and Review

A quadrilateral where the opposite sides are equal is a parallelogram. Justification: If opposite sides are equal (e.g., AB=CD and AD=BC), drawing a diagonal (say, BD) divides the quadrilateral into two triangles (∆ABD and ∆CDB). By SSS congruence, ∆ABD ≅ ∆CDB. This congruence implies that alternate interior angles are equal (e.g., ∠ABD = ∠CDB and ∠ADB = ∠CBD), which in turn proves that opposite sides are parallel (AB||CD and AD||BC). A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

True/False Statements:

• (i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
  • False [58(i)]. This description defines a rectangle. For it to be a square, the diagonals must also intersect at right angles.
• (ii) A quadrilateral having three right angles must be a rectangle.
  • True [58(ii)]. If three angles of a quadrilateral are 90°, then the sum of these angles is 270°. Since the total sum of angles in a quadrilateral is 360°, the fourth angle must also be 90° (360° - 270° = 90°). A quadrilateral with all four angles equal to 90° is a rectangle.
• (iii) A quadrilateral in which the opposite angles are equal must be a parallelogram.
  • True [59(v)]. In a quadrilateral, if opposite angles are equal, say ∠A=∠C and ∠B=∠D, then 2∠A + 2∠B = 360°, which means ∠A + ∠B = 180°. Since adjacent angles are supplementary, this implies that consecutive sides are parallel, thus forming a parallelogram.
• (iv) A quadrilateral in which all the angles are equal is a rectangle.
  • True [59(vi)]. If all four angles of a quadrilateral are equal and their sum is 360°, then each angle must be 360°/4 = 90°. A quadrilateral in which all angles are 90° is defined as a rectangle.

CLASS 8 NCERT WORKSHEET  CH-3 A STORY OF NUMBERS WITH ANSWER KEY

A Journey Through Numbers: Exploring Early Number Systems

Instructions: Answer the following questions based on the provided text, "A Story of Numbers." Ensure your answers are directly supported by the source material.

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Section 1: Foundations of Counting & Early Methods

1. When did humans first feel the need to count, and for what purposes?

   • Humans had the need to count as early as the Stone Age.
   • They counted to determine the quantity of food, the number of animals in their livestock, details regarding trades of goods, and the number of offerings given in rituals. They also wanted to keep track of passing days to predict important events like the new moon, full moon, or onset of a season.

2. What is a "one-to-one mapping" in the context of early counting? Provide an example from the text.

   • A one-to-one mapping is a way of associating each object to be counted with a distinct item, such as a stick or a sound, such that no two objects are associated or mapped to the same stick.
   • For example, early humans could keep one stick for every cow in a herd; the final collection of sticks would then represent the number of cows, which could be used to check if any cows were missing.

3. Name two of the oldest known mathematical artifacts featuring tally marks and state their approximate age and discovery location.

   • The Ishango bone, discovered in the Democratic Republic of Congo, dates back 20,000 to 35,000 years.
   • The Lebombo bone, discovered in South Africa, is an even older tally stick with 29 notches, estimated to be around 44,000 years old.

4. Describe the counting method of the Gumulgal people. What was a significant observation about their system compared to other cultures?

   • The Gumulgal, an indigenous group in Australia, formed their number names by counting in 2s. For instance, 3 was "ukasar-urapon" (2+1), and 4 was "ukasar-ukasar" (2+2). Any number greater than 6 was called "ras".
   • A significant and puzzling observation is that despite being geographically far apart and having no trace of contact, the Gumulgal, Bakairi (South America), and Bushmen (South Africa) developed equivalent number systems.

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Section 2: The Roman Numerals

1. List the "landmark numbers" and their associated symbols in the Roman numeral system up to 1,000.

   • I = 1
   • V = 5
   • X = 10
   • L = 50
   • C = 100
   • D = 500
   • M = 1,000

2. Represent the number 2367 in Roman numerals.

   • 2367 = 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1
   • In Roman numerals, this is MMCCCLXII.

3. What was a significant drawback of the Roman numeral system, especially for arithmetic operations?

   • Despite its relative efficiency, the Roman system does not lend itself to an easy performance of arithmetic operations, particularly multiplication and division. People using this system often relied on a calculating tool called the abacus for computations, which only specially trained people used.

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Section 3: The Idea of a Base & Egyptian System

1. Define a "base-n number system." What are its landmark numbers?

   • A base-n number system is a system where the first landmark number is 1 (n⁰), and every subsequent landmark number is obtained by multiplying the current landmark number by a fixed number 'n'.
   • Its landmark numbers are therefore the powers of n: n⁰, n¹, n², n³, and so on.

2. What is the base of the Egyptian number system? What are its landmark numbers?

   • The Egyptian number system is a base-10 system, also known as a decimal system.
   • Its landmark numbers are powers of 10: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 (10⁰, 10¹, 10², etc.).

3. Represent the number 324 in the Egyptian system using its principles (you can describe the symbols if you cannot draw them).

   • 324 = 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1.
   • In Egyptian numerals, this would be represented by three symbols for 100, two symbols for 10, and four symbols for 1. (The source image shows coiled ropes for 100, heel bones for 10, and single strokes for 1).

4. What is a significant advantage of a base-n system, particularly for arithmetic operations, compared to systems like Roman numerals?

   • The primary advantage is that it simplifies arithmetic operations.
   • Because the landmark numbers in a base-n system are powers of a single number (n), the product of any two landmark numbers is always another landmark number. This property significantly streamlines multiplication, unlike the Roman system where multiplication is difficult due to inconsistent grouping sizes for landmark numbers.

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Section 4: Place Value Representation & The Hindu Number System

1. Define a "positional number system" or "place value system."

   • A positional number system, also known as a place value system, is a number system with a base that uses the position of each symbol to determine the landmark number that it is associated with. This innovative idea allows for representing an unending sequence of numbers using only a finite number of different symbols.

2. Which ancient civilizations are mentioned as having used place value representations?

   • Mesopotamian (Babylonian)
   • Mayan
   • Chinese
   • Indian (Hindu)

3. What was the base of the Mesopotamian (Babylonian) number system? What was a key ambiguity in their early system, and how did later Mesopotamians attempt to address it?

   • The Mesopotamian system was a base-60 system, also called the sexagesimal system.
   • A key ambiguity was that a blank space was used to indicate the absence of a power of 60 in a position. This made it difficult to maintain consistent spacing and led to confusion, as the same numeral could be read in different ways.
   • To address this, later Mesopotamians used a "placeholder" symbol to denote a blank space, similar to our modern zero (0).

4. Describe the key features of the modern Hindu number system that make it highly efficient and widely used today.

   • It is a place value system.
   • It uses ten symbols (digits 0 through 9).
   • Crucially, it incorporates the digit 0 (zero), which functions both as a placeholder and as a number in its own right.
   • The use of 0 as a digit and the use of a single digit in each position ensures that this system does not lead to any ambiguity when reading or writing numerals.
   • This system allows for unambiguous representation of all numbers using a finite set of symbols and facilitates efficient computation.
   • It is considered one of the greatest, most creative, and most influential inventions of all time, forming the basis of much of modern science, technology, and mathematics.

5. Who were some key figures involved in the development and transmission of the Hindu number system, as mentioned in the text?

   • The Bakhshali manuscript (c. 3rd century CE) contains the first known instance of numbers written using ten digits, including 0.
   • Aryabhata (c. 499 CE) was the first mathematician to fully explain and perform elaborate scientific computations with the Indian system.
   • Al-Khwārizmī (c. 825 CE), a Persian mathematician, popularized the Hindu numerals in the Arab world through his book On the Calculation with Hindu Numerals.
   • Al-Kindi (c. 830 CE), a noted philosopher, also contributed to popularizing these numerals in the Arab world.
   • The Italian mathematician Fibonacci (c. 1200 CE) significantly advocated for Europe to adopt the Indian numerals.
   • Brahmagupta (628 CE) codified the use of 0 as a number like any other, on which basic arithmetic operations could be performed, laying foundations for modern mathematics.

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Section 5: Reflection & Comparison

1. Explain why the term "Hindu numerals" or "Indian numerals" is considered more accurate than "Arabic numerals" for the system we use today.

   • The modern number system originated and was developed in India around 2000 years ago.
   • While European scholars learned these numerals from the Arab world and called them "Arabic numerals," Arab scholars themselves, such as Al-Khwārizmī and Al-Kindi, referred to them as "Hindu numerals".
   • The term "Hindu" in this context refers to the geography and people from whom these numbers came, not a religion. The correction of this terminology is now occurring in many textbooks globally.

2. Based on the summary of ideas in number representation:

   • List the five key ideas in the evolution of number representation.

     1. Counting in groups of a single number (e.g., Gumulgal's system).
     2. Grouping using landmark numbers (e.g., Roman numerals).
     3. The idea of a base, by choosing powers of a number as landmark numbers (e.g., Egyptian system).
     4. The idea of a place value system, using positions to denote landmark numbers (e.g., Mesopotamian, Chinese).
     5. The idea of 0 as a positional digit and as a number (e.g., Hindu system).

   • Which of these ideas represents the "highest point" in the history of the evolution of number systems, and why?

     • The idea of place value marks the highest point in the history of the evolution of number systems.

When combined with the idea of 0 as a positional digit and as a number, it offers a remarkably elegant solution to the problem of representing the unending sequence of numbers using only a finite number of distinct symbols. Furthermore, it vastly simplifies computations and laid the foundations for modern mathematics and science.

A Journey Through Numbers: Exploring Early Number Systems - ANSWER KEY

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Section 1: Foundations of Counting & Early Methods

When did humans first feel the need to count, and for what purposes?

Humans had the need to count as early as the Stone Age.

They counted to determine the quantity of food, the number of animals in their livestock, details regarding trades of goods, and the number of offerings given in rituals. They also wanted to keep track of passing days to predict important events like the new moon, full moon, or onset of a season.

What is a "one-to-one mapping" in the context of early counting? Provide an example from the text.

A one-to-one mapping is a way of associating each object to be counted with a distinct item, such as a stick or a sound, such that no two objects are associated or mapped to the same stick.

For example, early humans could keep one stick for every cow in a herd; the final collection of sticks would then represent the number of cows, which could be used to check if any cows were missing.

Name two of the oldest known mathematical artifacts featuring tally marks and state their approximate age and discovery location.

The Ishango bone, discovered in the Democratic Republic of Congo, dates back 20,000 to 35,000 years.

The Lebombo bone, discovered in South Africa, is an even older tally stick with 29 notches, estimated to be around 44,000 years old.

Describe the counting method of the Gumulgal people. What was a significant observation about their system compared to other cultures?

The Gumulgal, an indigenous group in Australia, formed their number names by counting in 2s. For instance, 3 was "ukasar-urapon" (2+1), and 4 was "ukasar-ukasar" (2+2). Any number greater than 6 was called "ras".

A significant and puzzling observation is that despite being geographically far apart and having no trace of contact, the Gumulgal, Bakairi (South America), and Bushmen (South Africa) developed equivalent number systems.

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• Section 2: The Roman Numerals

List the "landmark numbers" and their associated symbols in the Roman numeral system up to 1,000.

I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1,000

Represent the number 2367 in Roman numerals.

2367 = 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1
In Roman numerals, this is MMCCCLXII.

What was a significant drawback of the Roman numeral system, especially for arithmetic operations?

Despite its relative efficiency, the Roman system does not lend itself to an easy performance of arithmetic operations, particularly multiplication and division. People using this system often relied on a calculating tool called the abacus for computations, which only specially trained people used.

• ---

• Section 3: The Idea of a Base & Egyptian System

Define a "base-n number system." What are its landmark numbers?

A base-n number system is a system where the first landmark number is 1 (n⁰), and every subsequent landmark number is obtained by multiplying the current landmark number by a fixed number 'n'.
Its landmark numbers are therefore the powers of n: n⁰, n¹, n², n³, and so on.

What is the base of the Egyptian number system? What are its landmark numbers?

The Egyptian number system is a base-10 system, also known as a decimal system.
Its landmark numbers are powers of 10: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 (10⁰, 10¹, 10², etc.).

Represent the number 324 in the Egyptian system using its principles (you can describe the symbols if you cannot draw them).

324 = 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1.
In Egyptian numerals, this would be represented by three symbols for 100 (coiled ropes), two symbols for 10 (heel bones), and four symbols for 1 (single strokes).

What is a significant advantage of a base-n system, particularly for arithmetic operations, compared to systems like Roman numerals?

The primary advantage is that it simplifies arithmetic operations.
Because the landmark numbers in a base-n system are powers of a single number (n), the product of any two landmark numbers is always another landmark number. This property significantly streamlines multiplication, unlike the Roman system where multiplication is difficult due to inconsistent grouping sizes for landmark numbers.

• ---

• Section 4: Place Value Representation & The Hindu Number System

Define a "positional number system" or "place value system."

A positional number system, also known as a place value system, is a number system with a base that uses the position of each symbol to determine the landmark number that it is associated with. This innovative idea allows for representing an unending sequence of numbers using only a finite number of different symbols.

Which ancient civilizations are mentioned as having used place value representations?

Mesopotamian (Babylonian)
Mayan
Chinese
Indian (Hindu)

What was the base of the Mesopotamian (Babylonian) number system? What was a key ambiguity in their early system, and how did later Mesopotamians attempt to address it?

The Mesopotamian system was a base-60 system, also called the sexagesimal system.
A key ambiguity was that a blank space was used to indicate the absence of a power of 60 in a position. This made it difficult to maintain consistent spacing and led to confusion, as the same numeral could be read in different ways (e.g., 60 and 3600 had ambiguous representations).
To address this, later Mesopotamians used a "placeholder" symbol to denote a blank space, similar to our modern zero (0).

Describe the key features of the modern Hindu number system that make it highly efficient and widely used today.

It is a place value system.
It uses ten symbols (digits 0 through 9).
Crucially, it incorporates the digit 0 (zero), which functions both as a placeholder and as a number in its own right.
The use of 0 as a digit and the use of a single digit in each position ensures that this system does not lead to any ambiguity when reading or writing numerals.
This system allows for unambiguous representation of all numbers using a finite set of symbols and facilitates efficient computation.
It is considered one of the greatest, most creative, and most influential inventions of all time, forming the basis of much of modern science, technology, and mathematics.

Who were some key figures involved in the development and transmission of the Hindu number system, as mentioned in the text?

The Bakhshali manuscript (c. 3rd century CE) contains the first known instance of numbers written using ten digits, including 0.
Aryabhata (c. 499 CE) was the first mathematician to fully explain and perform elaborate scientific computations with the Indian system.
Brahmagupta (628 CE) codified the use of 0 as a number like any other, on which basic arithmetic operations could be performed, laying foundations for modern mathematics.
Al-Khwārizmī (c. 825 CE), a Persian mathematician, popularized the Hindu numerals in the Arab world through his book On the Calculation with Hindu Numerals.
Al-Kindi (c. 830 CE), a noted philosopher, also contributed to popularizing these numerals in the Arab world.
The Italian mathematician Fibonacci (c. 1200 CE) significantly advocated for Europe to adopt the Indian numerals.

• ---

• Section 5: Reflection & Comparison

Explain why the term "Hindu numerals" or "Indian numerals" is considered more accurate than "Arabic numerals" for the system we use today.

The modern number system originated and was developed in India around 2000 years ago.
While European scholars learned these numerals from the Arab world and called them "Arabic numerals," Arab scholars themselves, such as Al-Khwārizmī and Al-Kindi, referred to them as "Hindu numerals".
The term "Hindu" in this context refers to the geography and people from whom these numbers came, not a religion. The correction of this terminology is now occurring in many textbooks globally.

Based on the summary of ideas in number representation:

• List the five key ideas in the evolution of number representation.

  • Counting in groups of a single number.
  • Grouping using landmark numbers.
  • The idea of a base, by choosing powers of a number as landmark numbers.
  • The idea of a place value system, using positions to denote landmark numbers.
  • The idea of 0 as a positional digit and as a number.

• Which of these ideas represents the "highest point" in the history of the evolution of number systems, and why?

  • The idea of place value marks the highest point in the history of the evolution of number systems.
  • When combined with the idea of 0 as a positional digit and as a number, it offers a remarkably elegant solution to the problem of representing the unending sequence of numbers using only a finite number of distinct symbols. Furthermore, it vastly simplifies computations and laid the foundations for modern mathematics and science.

CLASS 8 NCERT WORKSHEET CH-1 CUBES AND CUBEROOTS WITH ANSWER KEY

CLASS 8 NCERT WORKSHEET  CH-1 CUBES AND CUBE ROOTS

Worksheet: Exploring Cubes and Cube Roots

Instructions: Read each question carefully and use the information from the provided text to answer. Show your work where applicable.

Part A: Understanding Cubes

1. Define a "perfect cube" in your own words.

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2. How is the cube of any number 'n' denoted mathematically? n × n × n = __________

3. Complete the table of cubes for the given natural numbers:

   • 1³ = __________
   • 2³ = __________
   • 3³ = __________
   • 4³ = __________
   • 5³ = __________
   • 6³ = __________
   • 7³ = __________
   • 8³ = __________
   • 9³ = __________
   • 10³ = __________

4. Based on the cubes you filled in the table, what are the possible digits that a perfect cube can end with (its units place)?

   
   

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5. If a number contains zeros at the end, what can you say about the number of zeros its cube will have at the end? For example, can a cube end with exactly two zeros (00)? Explain.

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6. Explore the pattern showing the sum of consecutive odd numbers resulting in cube numbers:

   • 1 = 1 = 1³
   • 3 + 5 = __________ = 2³
   • 7 + 9 + 11 = __________ = 3³
   • 13 + 15 + 17 + 19 = __________ = __________
   • 21 + 23 + 25 + 27 + 29 = __________ = __________

Part B: Finding Cube Roots

1. Define "cube root" and state how it is denoted.

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2. How can prime factorization help in determining if a number is a perfect cube, and if so, in finding its cube root?

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3. Determine if the following numbers are perfect cubes using prime factorization. If they are, find their cube root.

   • (i) 64
     • Prime factorization of 64: ___________________________________
     • Can factors be grouped into triplets? (Yes/No): __________
     • Is 64 a perfect cube? __________
     • If yes, ³√64 = __________
   • (ii) 500
     • Prime factorization of 500: ___________________________________
     • Can factors be grouped into triplets? (Yes/No): __________
     • Is 500 a perfect cube? __________
     • If yes, ³√500 = __________

4. Find the cube roots of the following numbers using prime factorization:

   • (i) 27000
     • ³√27000 = __________
   • (ii) 10648
     • ³√10648 = __________

5. What number will you multiply by 1323 to make it a cube number? (Show your prime factorization and reasoning)

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6. Without using factorization, guess the cube roots of the following numbers based on patterns you might observe:

   • (i) 1331 (Hint: Consider what number's cube ends in 1)
     • ³√1331 = __________
   • (ii) 4913 (Hint: Consider what number's cube ends in 3)
     • ³√4913 = __________

Part C: True or False Statements

State whether the following statements are true or false. Explain your reasoning for each:

1. The cube of any odd number is even.

   • True / False: __________
   • Reasoning: _______________________________________________________________________________

2. There is no perfect cube that ends with 8.

   • True / False: __________
   • Reasoning: _________________________________________________________________________

3. The cube of a 2-digit number may be a 3-digit number.

   • True / False: __________
   • Reasoning: _________________________________________________________________________

4. The cube of a 2-digit number may have seven or more digits.

   • True / False: __________
   • Reasoning: _________________________________________________________________________

5. Cube numbers have an odd number of factors.

   • True / False: __________
   • Reasoning: _________________________________________________________________________

Part D: Challenge and Historical Context

1. Taxicab numbers are numbers that can be expressed as the sum of two cubes in two different ways. The number 1729 is the smallest taxicab number. How is it expressed as the sum of two cubes in two different ways?

   • 1729 = __________ + __________
   • 1729 = __________ + __________

2. In ancient Indian Sanskrit works, what term was used for a solid cube and also for the product of a number with itself three times? What term was used for the cube root?

   • Term for cube: __________
   • Term for cube root: __________
     
     
     

     Answer Key: Exploring Cubes and Cube Roots

     Part A: Understanding Cubes

     1. Define a "perfect cube" in your own words. A perfect cube is a number obtained by multiplying a number by itself three times. For example, 8 is a perfect cube because 2 × 2 × 2 = 8.

     2. How is the cube of any number 'n' denoted mathematically? n × n × n = n³.

     3. Complete the table of cubes for the given natural numbers:

        • 1³ = 1
        • 2³ = 8
        • 3³ = 27
        • 4³ = 64
        • 5³ = 125
        • 6³ = 216
        • 7³ = 343 (calculated from source 29, 17³=4913 ends in 3, 7³ ends in 3)
        • 8³ = 512
        • 9³ = 729
        • 10³ = 1000

     4. Based on the cubes you filled in the table, what are the possible digits that a perfect cube can end with (its units place)? By observing the cubes in the table, the possible units digits for perfect cubes are 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. (Unlike squares, cubes can end in any digit)

     5. If a number contains zeros at the end, what can you say about the number of zeros its cube will have at the end? For example, can a cube end with exactly two zeros (00)? Explain. If a number contains zeros at the end, its cube will have a number of zeros that is a multiple of three. For example, 10³ = 1000 (one zero becomes three), 20³ = 8000 (one zero becomes three), 100³ = 1,000,000 (two zeros become six). No, a cube cannot end with exactly two zeros (00) because the number of zeros at the end of a cube must be a multiple of three.

     6. Explore the pattern showing the sum of consecutive odd numbers resulting in cube numbers:

        • 1 = 1 = 1³
        • 3 + 5 = 8 = 2³
        • 7 + 9 + 11 = 27 = 3³
        • 13 + 15 + 17 + 19 = 64 = 4³
        • 21 + 23 + 25 + 27 + 29 = 125 = 5³

     Part B: Finding Cube Roots

     1. Define "cube root" and state how it is denoted. If y = x³, then x is the cube root of y. It is denoted by the symbol ³√.

     2. How can prime factorization help in determining if a number is a perfect cube, and if so, in finding its cube root? Prime factorization helps by allowing you to check if the prime factors of a number can be grouped into three identical groups (triplets). If they can, the number is a perfect cube. The cube root is then the product of the prime factors within one of these identical groups. Each prime factor of a number will appear three times in the prime factorization of its cube.

     3. Determine if the following numbers are perfect cubes using prime factorization. If they are, find their cube root.

        • (i) 64
          • Prime factorization of 64: 2 × 2 × 2 × 2 × 2 × 2 (or 2⁶)
          • Can factors be grouped into triplets? (Yes/No): Yes (2x2x2) x (2x2x2) or (2x2) x (2x2) x (2x2).
          • Is 64 a perfect cube? Yes.
          • If yes, ³√64 = 4 (since 4³=64).
        • (ii) 500
          • Prime factorization of 500: 2 × 2 × 5 × 5 × 5
          • Can factors be grouped into triplets? (Yes/No): No. The factor 2 appears only twice, not three times.
          • Is 500 a perfect cube? No.
          • If yes, ³√500 = N/A

     4. Find the cube roots of the following numbers using prime factorization:

        • (i) 27000
          • 27000 = 27 × 1000 = (3 × 3 × 3) × (10 × 10 × 10) = 3³ × 10³ = (3 × 10)³ = 30³
          • ³√27000 = 30
        • (ii) 10648
          • 10648 = 2 × 5324 = 2 × 2 × 2662 = 2 × 2 × 2 × 1331 = 2³ × 11³ = (2 × 11)³ = 22³
          • ³√10648 = 22

     5. What number will you multiply by 1323 to make it a cube number? (Show your prime factorization and reasoning) Prime factorization of 1323: 3 × 3 × 3 × 7 × 7 (or 3³ × 7²) To make it a perfect cube, each prime factor must appear in triplets. Here, 3 already appears as a triplet (3³), but 7 appears only twice (7²). Therefore, we need one more factor of 7 to complete the triplet for 7. So, you will multiply 1323 by 7.

     6. Without using factorization, guess the cube roots of the following numbers based on patterns you might observe:

        • (i) 1331
          • Hint: The number ends in 1. Looking at the cube table, numbers ending in 1 have cube roots ending in 1 (e.g., 1³=1, 11³=1331). Since 10³=1000, 1331 is likely 11³.
          • ³√1331 = 11
        • (ii) 4913
          • Hint: The number ends in 3. Looking at the cube table, numbers ending in 3 have cube roots ending in 7 (e.g., 7³=343, 17³=4913).
          • ³√4913 = 17

     Part C: True or False Statements

     State whether the following statements are true or false. Explain your reasoning for each:

     1. The cube of any odd number is even.

        • True / False: False
        • Reasoning: The cube of an odd number is always an odd number. For example, 1³ = 1, 3³ = 27, 5³ = 125, which are all odd.

     2. There is no perfect cube that ends with 8.

        • True / False: False
        • Reasoning: The number 8 itself is a perfect cube (2³ = 8). Also, 512 is a perfect cube (8³ = 512) and 12³ ends in 8.

     3. The cube of a 2-digit number may be a 3-digit number.

        • True / False: False
        • Reasoning: The smallest 2-digit number is 10. Its cube is 10³ = 1000, which is a 4-digit number. Any 2-digit number larger than 10 will have a cube greater than 1000.

     4. The cube of a 2-digit number may have seven or more digits.

        • True / False: False
        • Reasoning: The largest 2-digit number is 99. Its cube, 99³, is 970,299, which has 6 digits.

     5. Cube numbers have an odd number of factors.

        • True / False: False
        • Reasoning: Only square numbers have an odd number of factors. For a number to have an odd number of factors, it must have at least one factor that pairs with itself (e.g., 6x6 for 36). A number is a perfect cube if its prime factors can be split into three identical groups. A non-square cube, such as 8 (2x2x2), has factors 1, 2, 4, 8 (an even number of factors).

     Part D: Challenge and Historical Context

     1. Taxicab numbers are numbers that can be expressed as the sum of two cubes in two different ways. The number 1729 is the smallest taxicab number. How is it expressed as the sum of two cubes in two different ways?

        • 1729 = 1³ + 12³
        • 1729 = 9³ + 10³

     2. In ancient Indian Sanskrit works, what term was used for a solid cube and also for the product of a number with itself three times? What term was used for the cube root?

        • Term for cube: ghana
        • Term for cube root: ghana-mula

CLASS 8 NCERT WORKSHEET CH-2 Power Play WITH ANSWER KEY

 CLASS 8 NCERT WORKSHEET  CH-2 Power Play 

Class 8 - Chapter 2: Power Play Worksheet

Instructions: Read each question carefully and provide your answers based on the concepts discussed in the "Power Play" chapter. Show your work where necessary.

Section A: Understanding Exponential Growth

1. The Paper Folding Challenge:

   • According to Estu, what is the maximum number of times a sheet of paper can typically be folded?
   • Assume the initial thickness of a sheet of paper is 0.001 cm. If you could fold it as many times as you wish, what would its thickness be after:
     • 10 folds?
     • 17 folds?
     • 20 folds?
     • 30 folds?
   • The source states that after just 46 folds, the paper's thickness would reach the Moon! What is the estimated thickness after 46 folds mentioned in the text?
   • This rapid increase in thickness is an example of what type of growth?

2. Growth Analysis:

   • When a sheet of paper is folded, how does its thickness change after each fold?
   • By how many times does the thickness increase after any 3 folds? How about after any 10 folds?

Section B: Exponential Notation and Operations

1. Defining Exponents:

   • In the expression na, what is 'n' called, and what is 'a' called?
   • How is n2 read? How is n3 read?

2. Writing in Exponential Form:

   • Express the following in exponential form:
     • (i) 6 × 6 × 6 × 6
     • (ii) y × y
     • (iii) b × b × b × b
     • (iv) 5 × 5 × 7 × 7 × 7
     • (v) 2 × 2 × a × a
     • (vi) a × a × a × c × c × c × c × d

3. Calculating Numerical Values:

   • Write the numerical value of each of the following:
     • (i) 2 × 10^3
     • (ii) 7^2 × 2^3
     • (iv) (– 3)^2 × (– 5)^2

4. Prime Factorization:

   • Express the number 32400 as a product of its prime factors in exponential form.
   • Express each of the following as a product of powers of their prime factors in exponential form:
     • (i) 648
     • (ii) 405

5. The Stones that Shine:

   • In the "Stones that Shine" problem, how many rooms were there altogether, expressed in exponential form and as a numerical value?
   • How many diamonds were there in total, expressed in exponential form and as a numerical value?

Section C: Laws of Exponents

1. Multiplication Rule:

   • Generalize the product na × nb.
   • Use this rule to compute p^4 × p^6 in exponential form.

2. Power of a Power Rule:

   • Generalize the expression (na)b.
   • Write 8^6 as a power of a power in at least two different ways.

3. Combined Bases Rule:

   • Generalize the product ma × na.
   • Use this rule to compute the value of 2^5 × 5^5.

4. Division Rule:

   • Generalize the division na ÷ nb (where n ≠ 0 and a > b).
   • What is 2^100 ÷ 2^25 in powers of 2?
   • Why can't n be 0 in the division rule?

5. Zero Exponent:

   • What is the value of x^0 for any x ≠ 0? Provide a brief explanation.

6. Negative Exponents:

   • Generalize n–a and na in terms of negative exponents (where n ≠ 0).
   • Write equivalent forms of the following:
     • (i) 2–4
     • (ii) 10–5
     • (iii) (– 7)–2
   • Simplify and write the answers in exponential form:
     • (i) 2–4 × 27
     • (ii) 32 × 3–5 × 36
     • (iii) p3 × p–10

Section D: Scientific Notation (Standard Form)

1. Expanded Form with Powers of 10:

   • Write the number 47561 using powers of 10 in its expanded form.
   • Write 561.903 using powers of 10.

2. Converting to Scientific Notation:

   • What is the standard form of scientific notation defined as?
   • Express the following numbers in standard form:
     • (i) 59,853
     • (ii) 34,30,000
     • (iii) The distance between the Sun and Saturn: 14,33,50,00,00,000 m.
     • (iv) The mass of the Earth: 59,76,00,00,00,00,00,00,00,00,00,000 kg.

3. Comparing Large Numbers:

   • The distance between the Sun and Earth is 1.496 × 10^11 m. The distance between the Sun and Saturn is 1.4335 × 10^12 m. Which of these two distances is smaller? Explain your reasoning using scientific notation principles.
   • Explain why the exponent y in scientific notation (x × 10^y) is often considered more important than the coefficient x.

4. Real-World Applications of Scientific Notation:

   • Calculate and write the answer using scientific notation for the following:
     • (i) How many ants are there for every human in the world? (Global human population ≈ 8 × 10^9, global ant population ≈ 2 × 10^16)
     • (ii) If each tree had about 10^4 leaves, find the total number of leaves on all the trees in the world. (Estimated number of trees globally ≈ 3 × 10^12)
     • (iii) If one star is counted every second, how long would it take to count all the stars in the observable universe? (Estimated number of stars ≈ 2 × 10^23)
     • (iv) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees.

Section E: Problem-Solving and Concepts

1. Linear vs. Exponential Growth:

   • Explain the difference between linear growth and exponential growth. Give an example of each.
   • To reach the Moon (3,84,400 km) using a ladder with steps 20 cm apart (linear growth) takes approximately how many steps?
   • Compare this to the number of paper folds (exponential growth) needed to reach the Moon.

2. Combinations:

   • Estu has 4 dresses and 3 caps. How many different combinations are possible?
   • A safe is secured with a 5-digit password. How many possible passwords are there if each digit can be from 0 to 9? Express your answer using exponential notation.
   • If Estu buys a lock with 6 slots using letters A to Z, how many passwords are possible? (Assume 26 options for each slot).

3. Estimating Time:

   • If you have lived for a million seconds (10^6 seconds), approximately how old would you be in days?
   • How long ago did plants on land start to appear? Express your answer in seconds using scientific notation. (Hint: Plants on land started 47 crore years ago)

4. Summary of Key Rules:

   • Write down the six generalized forms for operations with exponents mentioned in the summary.
     
     
     

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     Class 8 - Chapter 2: Power Play Worksheet - Answer Key

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     Section A: Understanding Exponential Growth

     1. The Paper Folding Challenge:

        • According to Estu, a sheet of paper can’t be folded more than 7 times.
        • Assume the initial thickness of a sheet of paper is 0.001 cm.
          • After 10 folds: 1.024 cm.
          • After 17 folds: ≈ 131 cm.
          • After 20 folds: ≈ 10.4 m.
          • After 30 folds: about 10.7 km.
        • The source states that after just 46 folds, the paper's thickness would reach the Moon! The estimated thickness after 46 folds is more than 7,00,000 km.
        • This rapid increase in thickness is an example of multiplicative growth, also called exponential growth.

     2. Growth Analysis:

        • When a sheet of paper is folded, its thickness doubles after each fold.
        • After any 3 folds, the thickness increases 8 times (= 2 × 2 × 2).
        • After any 10 folds, the thickness increases by 1024 times (= 2 multiplied by itself 10 times).

     Section B: Exponential Notation and Operations

     1. Defining Exponents:

        • In the expression na, 'n' is the base and 'a' is the exponent/power.
        • n2 is read as ‘n squared’ or ‘n raised to the power 2’. n3 is read as ‘n cubed’ or ‘n raised to the power 3’.

     2. Writing in Exponential Form:

        • (i) 6 × 6 × 6 × 6: 6^4
        • (ii) y × y: y^2
        • (iii) b × b × b × b: b^4
        • (iv) 5 × 5 × 7 × 7 × 7: 5^2 × 7^3 (Based on examples like a3b2)
        • (v) 2 × 2 × a × a: 2^2 × a^2 (Based on examples like a3b2)
        • (vi) a × a × a × c × c × c × c × d: a^3 × c^4 × d

     3. Calculating Numerical Values:

        • (i) 2 × 10^3: 2 × 1000 = 2000
        • (ii) 7^2 × 2^3: 49 × 8 = 392
        • (iv) (– 3)^2 × (– 5)^2: 9 × 25 = 225

     4. Prime Factorization:

        • The number 32400 as a product of its prime factors in exponential form: 32400 = 2^4 × 5^2 × 3^4.
        • Express each of the following as a product of powers of their prime factors in exponential form:
          • (i) 648: To find this, we would factorize 648. 648 = 2 × 324 = 2 × 2 × 162 = 2 × 2 × 2 × 81 = 2^3 × 3^4. So, 648 = 2^3 × 3^4 (Concept applied from example)
          • (ii) 405: To find this, we would factorize 405. 405 = 5 × 81 = 5 × 3 × 27 = 5 × 3 × 3 × 9 = 5 × 3 × 3 × 3 × 3 = 5 × 3^4. So, 405 = 5 × 3^4 (Concept applied from example)

     5. The Stones that Shine:

        • There were altogether 3^4 rooms, which is 81 rooms.
        • There were in total 3^7 diamonds, which is 2187 diamonds.

     Section C: Laws of Exponents

     1. Multiplication Rule:

        • The generalized product na × nb is na+b.
        • Using this rule, p^4 × p^6 in exponential form is p^(4+6) = p^10.

     2. Power of a Power Rule:

        • The generalized expression (na)b is (nb)a = na × b = nab.
        • Write 8^6 as a power of a power in at least two different ways:
          • 8^6 = (8^2)^3 (since 2 × 3 = 6)
          • 8^6 = (8^3)^2 (since 3 × 2 = 6)

     3. Combined Bases Rule:

        • The generalized product ma × na is (mn)a.
        • Using this rule, the value of 2^5 × 5^5 is (2 × 5)^5 = 10^5 = 1,00,000.

     4. Division Rule:

        • The generalized division na ÷ nb (where n ≠ 0 and a > b) is na – b.
        • 2^100 ÷ 2^25 in powers of 2 is 2^(100 – 25) = 2^75.
        • n cannot be 0 in the division rule because division by zero is undefined. If n were 0, na ÷ nb would involve division by zero.

     5. Zero Exponent:

        • The value of x^0 for any x ≠ 0 is 1.
        • This is because x^0 can be thought of as xa ÷ xa. Since any non-zero number divided by itself is 1, xa ÷ xa = 1. Therefore, x^0 = 1.

     6. Negative Exponents:

        • The generalized forms for negative exponents (where n ≠ 0) are n–a = 1/na and na = 1/n–a.
        • Write equivalent forms of the following:
          • (i) 2–4: 1/2^4
          • (ii) 10–5: 1/10^5
          • (iii) (– 7)–2: 1/(–7)^2
        • Simplify and write the answers in exponential form: (Applying na × nb = na+b and na ÷ nb = na – b and similar rules for negative exponents where a and b can be any integers)
          • (i) 2–4 × 27: 2^(–4 + 7) = 2^3
          • (ii) 32 × 3–5 × 36: 3^(2 + (–5) + 6) = 3^(2 – 5 + 6) = 3^3
          • (iii) p3 × p–10: p^(3 + (–10)) = p^(3 – 10) = p–7

     Section D: Scientific Notation (Standard Form)

     1. Expanded Form with Powers of 10:

        • 47561: (4 × 10^4) + (7 × 10^3) + (5 × 10^2) + (6 × 10^1) + (1 × 10^0).
        • 561.903: (5 × 10^2) + (6 × 10^1) + (1 × 10^0) + (9 × 10–1) + (0 × 10–2) + (3 × 10–3).

     2. Converting to Scientific Notation:

        • The standard form of scientific notation (also called standard form) is written as x × 10^y, where x ≥ 1 and x < 10 is the coefficient and y, the exponent, is any integer.
        • Express the following numbers in standard form:
          • (i) 59,853: 5.9853 × 10^4 (Concept applied from example)
          • (ii) 34,30,000: 3.43 × 10^6 (Concept applied from example)
          • (iii) The distance between the Sun and Saturn: 1.4335 × 10^12 m.
          • (iv) The mass of the Earth: 5.976 × 10^24 kg (Calculation: moving the decimal point 24 places to the left to get a number between 1 and 10).

     3. Comparing Large Numbers:

        • The distance between the Sun and Earth is 1.496 × 10^11 m. The distance between the Sun and Saturn is 1.4335 × 10^12 m.
          • The distance between the Sun and Earth (1.496 × 10^11 m) is smaller.
          • Reasoning: When comparing numbers in scientific notation, first compare the exponents of 10. 10^11 is smaller than 10^12. Therefore, 1.496 × 10^11 m is smaller than 1.4335 × 10^12 m.
        • The exponent y in scientific notation (x × 10^y) is often considered more important than the coefficient x because the exponent indicates the number of digits or the order of magnitude of the number. For example, changing the exponent y by 1 changes the number by 10 times, whereas changing the coefficient x only changes it proportionally (e.g., from 2 crore to 3 crore vs. 2 crore to 20 crore).

     4. Real-World Applications of Scientific Notation:

        • (i) How many ants are there for every human in the world?
          • Global human population ≈ 8 × 10^9.
          • Global ant population ≈ 2 × 10^16.
          • Ants per human = (2 × 10^16) ÷ (8 × 10^9) = (2/8) × 10^(16-9) = 0.25 × 10^7 = 2.5 × 10^6 ants per human.
        • (ii) If each tree had about 10^4 leaves, find the total number of leaves on all the trees in the world.
          • Estimated number of trees globally ≈ 3 × 10^12.
          • Total leaves = (3 × 10^12) × 10^4 = 3 × 10^(12+4) = 3 × 10^16 leaves.
        • (iii) If one star is counted every second, how long would it take to count all the stars in the observable universe?
          • Estimated number of stars ≈ 2 × 10^23.
          • Time to count = 2 × 10^23 seconds.
          • This is a very long time, equivalent to 2 × 10^23 seconds.
        • (iv) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees.
          • 100 million = 1 × 10^8 colonies.
          • 50,000 = 5 × 10^4 bees per colony.
          • Total honeybees = (1 × 10^8) × (5 × 10^4) = 5 × 10^(8+4) = 5 × 10^12 bees.

     Section E: Problem-Solving and Concepts

     1. Linear vs. Exponential Growth:

        • Linear growth involves a fixed increase (additive) in quantity over time, whereas exponential growth involves a multiplicative increase where the quantity doubles or triples (multiplies by a constant factor) over time.
        • Example of linear growth: Climbing a ladder where each step increases height by a fixed distance (e.g., 20 cm).
        • Example of exponential growth: The thickness of a paper doubling with each fold, the number of lotuses doubling in a pond, or the number of combinations in a password.
        • To reach the Moon (3,84,400 km) using a ladder with steps 20 cm apart (linear growth) takes approximately 1,92,20,00,000 steps.
        • To reach the Moon using paper folds (exponential growth) takes just 46 folds.

     2. Combinations:

        • Estu has 4 dresses and 3 caps. The number of different combinations possible is 4 × 3 = 12 combinations.
        • A safe is secured with a 5-digit password. Each digit can be from 0 to 9 (10 options). The number of possible passwords is 10 × 10 × 10 × 10 × 10 = 10^5 = 1,00,000 passwords.
        • If Estu buys a lock with 6 slots using letters A to Z (26 options for each slot), the number of possible passwords is 26 × 26 × 26 × 26 × 26 × 26 = 26^6 (Based on the password example).

     3. Estimating Time:

        • If you have lived for a million seconds (10^6 seconds), you would be approximately 11.57 days old.
        • Plants on land started 47 crore years ago. 47 crore years = 470 million years.
          • The source gives 10^16 seconds ≈ 31.7 crore years.
          • Using this, 47 crore years is approximately (47/31.7) × 10^16 seconds ≈ 1.48 × 10^16 seconds.

     4. Summary of Key Rules:

        • The six generalized forms for operations with exponents are:
          • na × nb = na+b
          • (na)b = (nb)a = na × b
          • na ÷ nb = na – b (n ≠ 0)
          • na × ma = (n × m)a
          • na ÷ ma = (n ÷ m)a (m ≠ 0)
          • n0 = 1 (n ≠ 0)