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.

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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)

CLASS 8 NCERT WORKSHEET CH-1 squares and square roots WITH ANSWERKEY

CLASS 8 WORKSHEET 
CH-1 squares and square roots

Worksheet: Exploring Squares and Square Roots

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

Part A: Understanding Squares

1. Define a "square number" or "perfect square" in your own words.

   
   

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2. The area of a square is found by multiplying its side length by itself.

• If a square has a side length of 5 units, what is its area? Express this using square notation. Area = _______________________
• What notation do we use for any number 'n' multiplied by itself? n × n = __________

3. List all the perfect squares between 1 and 100.

   
   

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Part B: Properties of Perfect Squares

1. What are the only possible digits that a perfect square can end with (its units place)?

   
   

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2. Which of the following numbers are not perfect squares? Explain your reasoning based on their unit digits.

   • (i) 2032
   • (ii) 1027
   • (iii) 345
   • (iv) 576
   • (v) 2048
   • (vi) 1089

   Not Perfect Squares: ____________________________________________________  Reasoning:____________________________________________________________________

   
   

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3. Observe the pattern for the number of zeros at the end of a number and its square.

   • If a number ends with three zeros (e.g., 1000), how many zeros will its square have at the end?

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   • What can you conclude about the number of zeros at the end of any perfect square?

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4. Complete the following pattern showing the sum of consecutive odd numbers resulting in square numbers:

   • 1 = 1 = 1²
   • 1 + 3 = 4 = 2²
   • 1 + 3 + 5 = 9 = 3²
   • 1 + 3 + 5 + 7 = __________ = __________
   • 1 + 3 + 5 + 7 + 9 = __________ = __________
   • 1 + 3 + 5 + 7 + 9 + 11 = __________ = __________

5. Given that 35² = 1225, use the pattern of adding consecutive odd numbers to find 36².

   • What is the 36th odd number? (Hint: The nth odd number is 2n-1)

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   • Calculate 36²: 36² = 35² + (36th odd number) = _______________________

6. How many numbers lie between the squares of 16 and 17?

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Part C: Finding Square Roots

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

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2. Every perfect square has two integer square roots. What are they for the number 64?

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3. Determine if 1156 is a perfect square using prime factorization. If it is, find its square root. (Show your prime factorization steps)

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   Is 1156 a perfect square? __________ If yes, √1156 = __________

4. Find the square root of 441 m² (area of a square) to determine the length of its side. Length of side = _______________________
   
   
   

5. Estimate the square root of 250 without calculating it exactly. Explain your reasoning by identifying the closest perfect squares.

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6. Akhil has a square piece of cloth with an area of 125 cm². He wants to cut out the largest possible square handkerchief with an integer side length. What is the maximum side length he can cut?

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Part D: Challenge Question

1. Recall the puzzle given by Queen Ratnamanjuri involving 100 lockers.

   • Person 1 opens every locker.
   • Person 2 toggles every 2nd locker.
   • Person 3 toggles every 3rd locker, and so on, until all 100 people have taken their turn.

   Khoisnam immediately knew which lockers would remain open at the end. How did he know? (Hint: Think about how many times each locker is toggled and what type of numbers have an odd number of factors).

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   Which specific locker numbers (up to 100) would remain open?

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Answer Key: Exploring Squares and Square Roots

Part A: Understanding Squares

1. Define a "square number" or "perfect square" in your own words. A square number, or perfect square, is a number obtained by multiplying a number by itself. It can also be described as the area of a square figure. Example definition: A perfect square is a number that results from multiplying an integer by itself. For instance, 4 is a perfect square because 2 × 2 = 4.

2. The area of a square is found by multiplying its side length by itself.

   • If a square has a side length of 5 units, what is its area? Express this using square notation. Area = 5 × 5 = 25 square units, or 5² = 25.
   • What notation do we use for any number 'n' multiplied by itself? n × n = n².

3. List all the perfect squares between 1 and 100. The perfect squares between 1 and 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Part B: Properties of Perfect Squares

1. What are the only possible digits that a perfect square can end with (its units place)? Perfect squares can only end with the digits 0, 1, 4, 5, 6, or 9.

2. Which of the following numbers are not perfect squares? Explain your reasoning based on their unit digits.

   • (i) 2032
   • (ii) 1027
   • (iii) 345
   • (iv) 576
   • (v) 2048
   • (vi) 1089

   Not Perfect Squares: (i) 2032, (ii) 1027, (v) 2048. Reasoning: If a number ends with 2, 3, 7, or 8, then it is definitely not a square.

3. Observe the pattern for the number of zeros at the end of a number and its square.

   • If a number ends with three zeros (e.g., 1000), how many zeros will its square have at the end? Six zeros (e.g., 1000² = 1,000,000).
   • What can you conclude about the number of zeros at the end of any perfect square? Perfect squares can only have an even number of zeros at the end.

4. Complete the following pattern showing the sum of consecutive odd numbers resulting in square numbers:

   • 1 = 1 = 1²
   • 1 + 3 = 4 = 2²
   • 1 + 3 + 5 = 9 = 3²
   • 1 + 3 + 5 + 7 = 16 = 4²
   • 1 + 3 + 5 + 7 + 9 = 25 = 5²
   • 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²

5. Given that 35² = 1225, use the pattern of adding consecutive odd numbers to find 36².

   • What is the 36th odd number? (Hint: The nth odd number is 2n-1) The nth odd number is 2n–1. So, the 36th odd number is (2 × 36) - 1 = 71.
   • Calculate 36²: 36² = 35² + (36th odd number) = 1225 + 71 = 1296.

6. How many numbers lie between the squares of 16 and 17? This information is not directly provided in the source material in an explicit statement. However, based on the pattern n² and (n+1)², the numbers between them are (n+1)² - n² - 1 = (n² + 2n + 1) - n² - 1 = 2n. So, between 16² and 17² (where n=16), there are 2 × 16 = 32 numbers [16 (Implied by the question type)].

Part C: Finding Square Roots

1. Define "square root" and state how it is denoted. If y = x², then x is the square root of y. It is the inverse operation of squaring a number. The square root of a number is denoted by the symbol √.

2. Every perfect square has two integer square roots. What are they for the number 64? The square roots of 64 are +8 and –8.

3. Determine if 1156 is a perfect square using prime factorization. If it is, find its square root. (Show your prime factorization steps) Prime factorization of 1156: 1156 = 2 × 578 578 = 2 × 289 289 = 17 × 17 So, 1156 = 2 × 2 × 17 × 17.

   • Can factors be grouped into pairs? Yes, (2 × 2) × (17 × 17). This can be grouped as (2 × 17) × (2 × 17).
   • Is 1156 a perfect square? Yes.
   • If yes, √1156 = 2 × 17 = 34.

4. Find the square root of 441 m² (area of a square) to determine the length of its side. Length of side = √441 = 21 m.

5. Estimate the square root of 250 without calculating it exactly. Explain your reasoning by identifying the closest perfect squares. We know that 15² = 225 and 16² = 256. Therefore, the square root of 250 is between 15 and 16. Since 256 is much closer to 250 than 225, the square root of 250 is approximately 16 (but less than 16).

6. Akhil has a square piece of cloth with an area of 125 cm². He wants to cut out the largest possible square handkerchief with an integer side length. What is the maximum side length he can cut? The nearest perfect square less than 125 is 121 (which is 11²). Therefore, the maximum side length he can cut with an integer side length is 11 cm.

Part D: Challenge Question

1. Recall the puzzle given by Queen Ratnamanjuri involving 100 lockers.

   • Person 1 opens every locker.
   • Person 2 toggles every 2nd locker.
   • Person 3 toggles every 3rd locker, and so on, until all 100 people have taken their turn.

   Khoisnam immediately knew which lockers would remain open at the end. How did he know? (Hint: Think about how many times each locker is toggled and what type of numbers have an odd number of factors). Khoisnam knew that a locker would remain open only if it was toggled an odd number of times. The number of times a locker is toggled is equal to the number of its factors. Most numbers have an even number of factors because factors usually come in pairs (e.g., for 6, factors are 1&6, 2&3). However, numbers that are perfect squares have an odd number of factors because one factor is paired with itself (e.g., for 36, 6 × 6). Thus, every locker whose number is a perfect square will remain open.

   Which specific locker numbers (up to 100) would remain open? The locker numbers that remain open are the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.