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.
     
     
     

     ---

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

     ---

     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)