Chapter 2 exercise answers: Power Play

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(i) 6 × 6 × 6 × 6

1. Count how many times 6 is multiplied: 4 times.
2. Write it as a power: 6⁴.

Answer6⁴

(ii) y × y

1. y is multiplied by itself 2 times.
2. Write it as y².

Answery²

(iii) b × b × b × b

1. b is multiplied by itself 4 times.
2. Write it as b⁴.

Answerb⁴

(iv) 5 × 5 × 7 × 7 × 7

1. Count the 5's: 2 of them, so 5².
2. Count the 7's: 3 of them, so 7³.
3. Multiply the two powers: 5² × 7³.

Answer5² × 7³

(v) 2 × 2 × a × a

1. Count the 2's: 2 of them, so 2².
2. Count the a's: 2 of them, so a².
3. Multiply the two powers: 2² × a².

Answer2² × a²

(vi) a × a × a × c × c × c × c × d

1. Count the a's: 3 of them, so a³.
2. Count the c's: 4 of them, so c⁴.
3. d appears only once, so it stays as d.
4. Multiply the three powers: a³ × c⁴ × d.

Answera³ × c⁴ × d

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(i) 648

1. Divide 648 by 2 repeatedly: 648 → 324 → 162 → 81 — that's three 2's.
2. 81 is 3⁴.
3. So 648 = 2³ × 3⁴.

Answer2³ × 3⁴

(ii) 405

1. Divide 405 by 5 once: 405 → 81.
2. 81 is 3⁴.
3. So 405 = 3⁴ × 5.

Answer3⁴ × 5

(iii) 540

1. Divide 540 by 2 twice: 540 → 270 → 135.
2. Divide 135 by 3 three times: 135 → 45 → 15 → 5.
3. 5 is left over.
4. So 540 = 2² × 3³ × 5.

Answer2² × 3³ × 5

(iv) 3600

1. Divide 3600 by 2 four times: 3600 → 1800 → 900 → 450 → 225.
2. Divide 225 by 3 twice: 225 → 75 → 25.
3. 25 is 5².
4. So 3600 = 2⁴ × 3² × 5².

Answer2⁴ × 3² × 5²

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(i) 2 × 10³

1. 10³ = 1000.
2. 2 × 1000 = 2000.

Answer2000

(ii) 7² × 2³

1. 7² = 49.
2. 2³ = 8.
3. 49 × 8 = 392.

Answer392

(iii) 3 × 4⁴

1. 4⁴ = 256.
2. 3 × 256 = 768.

Answer768

(iv) (−3)² × (−5)²

1. (−3)² = 9 (an even power of a negative number is positive).
2. (−5)² = 25.
3. 9 × 25 = 225.

Answer225

(v) 3² × 10⁴

1. 3² = 9.
2. 10⁴ = 10000.
3. 9 × 10000 = 90000.

Answer90000

(vi) (−2)⁵ × (−10)⁶

1. (−2)⁵ = −32 (an odd power of a negative number is negative).
2. (−10)⁶ = 1000000 (an even power of a negative number is positive).
3. −32 × 1000000 = −32000000.

Answer−32000000

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Open NCERT p. 44Matches NCERT’s answer

1. Rewrite 4 as 2², so 4³² = (2²)³² = 2⁶⁴.
2. Now 2²²⁴ ÷ 2⁶⁴ = 2 to the power (224 − 64) = 2¹⁶⁰.
3. Units digits of powers of 2 repeat in a cycle of 4: 2, 4, 8, 6, 2, 4, 8, 6, …
4. 160 is exactly divisible by 4, so 2¹⁶⁰ ends in the same digit as 2⁴, which is 6.

Answer6

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Open NCERT p. 44Matches NCERT’s answer

1. Each container always has 5 bottles — the number of bottles in one container never changes.
2. One new container arrives each day, so after 40 days there are 40 containers in all.
3. Total bottles = 5 × 40 = 200.

Answer200

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Open NCERT p. 44Checked by computerAnswers can differ: one example

(i) 64³

1. 64 = 2⁶, so 64³ = 2^{6 × 3} = 2¹⁸.
2. Way 1: split the exponent 18 as 9 + 9, giving 2⁹ × 2⁹.
3. Way 2: 2¹⁸ = (2²)⁹ = 4⁹. Split 9 as 5 + 4, giving 4⁵ × 4⁴.
4. Way 3: exponents may be negative, and 20 + (−2) = 18, giving 2²⁰ × 2⁻².
5. Many other answers are also correct; any product whose exponents give 2¹⁸ will do.

Answer2⁹ × 2⁹, 4⁵ × 4⁴, 2²⁰ × 2⁻²

(ii) 192⁸

1. 192 = 64 × 3 = 2⁶ × 3, so 192⁸ = 2^{6 × 8} × 3⁸ = 2⁴⁸ × 3⁸.
2. Way 1: 2⁴⁸ × 3⁸.
3. Way 2: 2⁴⁸ = (2²)²⁴ = 4²⁴, giving 4²⁴ × 3⁸.
4. Way 3: 2⁴⁸ = (2⁴)¹² = 16¹², giving 16¹² × 3⁸.
5. Other splits are also correct, for example 2⁴⁰ × 2⁸ × 3⁸.

Answer2⁴⁸ × 3⁸, 4²⁴ × 3⁸, 16¹² × 3⁸

(iii) 32⁻⁵

1. 32 = 2⁵, so 32⁻⁵ = 2^{5 × (−5)} = 2⁻²⁵.
2. Way 1: (−13) + (−12) = −25, giving 2⁻¹³ × 2⁻¹².
3. Way 2: 4⁻¹³ = (2²)⁻¹³ = 2⁻²⁶, and −26 + 1 = −25, giving 4⁻¹³ × 2¹.
4. Way 3: 8⁻⁹ = (2³)⁻⁹ = 2⁻²⁷, and −27 + 2 = −25, giving 8⁻⁹ × 2².
5. Other splits are also correct, for example 2⁻¹⁰ × 2⁻¹⁵.

Answer2⁻¹³ × 2⁻¹², 4⁻¹³ × 2¹, 8⁻⁹ × 2²

Watch this explained “Cut it somewhere else”, 1:27 into The laws of exponents, and where each one comes from · हिंदी में देखें

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(i) Cube numbers are also square numbers.

1. 1 is a cube (1³) and also a square (1²), so the statement can be true.
2. 8 is a cube (2³) but not a square number (no whole number squares to give 8), so it can be false too.
3. It is true for some numbers and false for others.

AnswerOnly Sometimes True

(ii) Fourth powers are also square numbers.

1. A fourth power is n⁴ for some whole number n.
2. n⁴ = (n²)², which is a whole number squared.
3. Every fourth power is a square number.

AnswerAlways True

(iii) The fifth power of a number is divisible by the cube …

1. The fifth power of a number n is n⁵.
2. n⁵ = n² × n³, so n⁵ ÷ n³ = n², always a whole number.
3. n⁵ is always divisible by n³.

AnswerAlways True

(iv) The product of two cube numbers is a cube number.

1. Take two cube numbers, a³ and b³.
2. Their product is a³ × b³ = (a × b)³, which is a cube number.
3. This works for any a and b.

AnswerAlways True

(v) q⁴⁶ is both a 4th power and a 6th power

1. q⁴⁶ is a perfect 4th power only if 46 splits into equal groups of 4, i.e. only if 4 divides 46.
2. 46 ÷ 4 leaves a remainder, so q⁴⁶ is never a perfect 4th power.
3. 46 ÷ 6 also leaves a remainder, so q⁴⁶ is never a perfect 6th power either.

AnswerNever True

Watch the lesson The laws of exponents, and where each one comes from · हिंदी में देखें

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(i) 10⁻² × 10⁻⁵

1. Same base, multiplying: add the exponents.
2. −2 + (−5) = −7.

Answer10⁻⁷

(ii) 5⁷ ÷ 5⁴

1. Same base, dividing: subtract the exponents.
2. 7 − 4 = 3.

Answer5³

(iii) 9⁻⁷ ÷ 9⁴

1. Same base, dividing: subtract the exponents.
2. −7 − 4 = −11.

Answer9⁻¹¹

(iv) (13⁻²)⁻³

1. A power raised to a power: multiply the exponents.
2. −2 × −3 = 6.

Answer13⁶

(v) m⁵n¹²(mn)⁹

1. (mn)⁹ = m⁹n⁹ (each factor inside the bracket gets the power).
2. Combine the m's: m⁵ × m⁹ = m¹⁴.
3. Combine the n's: n¹² × n⁹ = n²¹.

Answerm¹⁴n²¹

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(i) (1.2)²

1. 1.2 is 12 divided by 10.
2. Squaring divides the answer by 10² = 100.
3. 144 ÷ 100 = 1.44.

Answer1.44

(ii) (0.12)²

1. 0.12 is 12 divided by 100.
2. Squaring divides the answer by 100² = 10000.
3. 144 ÷ 10000 = 0.0144.

Answer0.0144

(iii) (0.012)²

1. 0.012 is 12 divided by 1000.
2. Squaring divides the answer by 1000² = 1000000.
3. 144 ÷ 1000000 = 0.000144.

Answer0.000144

(iv) 120²

1. 120 is 12 multiplied by 10.
2. Squaring multiplies the answer by 10² = 100.
3. 144 × 100 = 14400.

Answer14400

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1. Label the five expressions in order A to E: A = 2⁴ × 3⁶, B = 6⁴ × 3², C = 6¹⁰, D = 18² × 6², E = 6²⁴.
2. B: 6⁴ × 3² = (2 × 3)⁴ × 3² = 2⁴ × 3⁴ × 3² = 2⁴ × 3⁶ — the same as A.
3. D: 18² × 6² = (18 × 6)² = 108² = 11664, and A = 2⁴ × 3⁶ = 16 × 729 = 11664 too — the same as A.
4. C = 6¹⁰ and E = 6²⁴ are much bigger powers of 6 and do not equal 11664.
5. So A, B and D are all equal to 11664.

AnswerA, B and D — that is, 2⁴ × 3⁶, 6⁴ × 3² and 18² × 6² — are all equal to 11664.

Watch this explained “The third law, and its twin”, 7:36 into The laws of exponents, and where each one comes from · हिंदी में देखें

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(i) 4³ or 3⁴

1. 4³ = 4 × 4 × 4 = 64.
2. 3⁴ = 3 × 3 × 3 × 3 = 81.
3. 81 is greater than 64.

Answer3⁴ (= 81) is greater than 4³ (= 64)

(ii) 2⁸ or 8²

1. 2⁸ = 256.
2. 8² = 64.
3. 256 is greater than 64.

Answer2⁸ (= 256) is greater than 8² (= 64)

(iii) 100² or 2¹⁰⁰

1. 100² = 10000.
2. Keep doubling from 2¹⁰ = 1024: 2¹¹ = 2048, 2¹² = 4096, 2¹³ = 8192, 2¹⁴ = 16384. Already 2¹⁴ is more than 10000.
3. 2¹⁰⁰ is 2¹⁴ multiplied by 2 another 86 times, so it is far greater than 10000.

Answer2¹⁰⁰ is greater than 100² (= 10000)

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1. With d digits (each 0–9), the number of different codes possible is 10 to the power d.
2. 9 digits give 10⁹ = 1,000,000,000 codes, which is 1 billion — fewer than the 8.5 billion packets needed.
3. 10 digits give 10¹⁰ = 10,000,000,000 codes, which is 10 billion — enough for 8.5 billion packets.
4. So the code must have 10 digits.

Answer10

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1. Write the number as a product of prime powers. It is a square exactly when every prime's exponent is even, and a cube exactly when every exponent is a multiple of 3.
2. To be both, every exponent must be a multiple of 2 and of 3, so a multiple of 6. That means the number is a 6th power, n⁶.
3. Every 6th power works: n⁶ = (n³)² is a square and n⁶ = (n²)³ is a cube.
4. Examples: 1⁶ = 1, 2⁶ = 64 = 8² = 4³, 3⁶ = 729 = 27² = 9³, 4⁶ = 4096 = 64² = 16³.
5. There is one for every n, so there are infinitely many.

AnswerYes. The numbers that are both squares and cubes are exactly the 6th powers, n⁶: 1, 64, 729, 4096, …

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1. Each of the 5 slots can be filled by any of the 10 digits (0–9) or 26 letters (A–Z): 10 + 26 = 36 choices.
2. The slots are filled independently, so multiply the choices for each slot: 36 × 36 × 36 × 36 × 36 = 36⁵.
3. 36⁵ = 60,466,176.

Answer36⁵ = 60,466,176

Watch this explained “The lock”, 8:18 into Exponential notation: repeated multiplication written once · हिंदी में देखें

Open NCERT p. 45Matches NCERT’s answer

1. Sheep ≈ 10⁹ and goats ≈ 10⁹, so the total is 10⁹ + 10⁹.
2. Two equal amounts added make twice that amount: 10⁹ + 10⁹ = 2 × 10⁹ (2 billion).
3. Adding does not add or multiply the exponents: 10¹⁸ = 10⁹ × 10⁹ would be for multiplying, and 10¹⁰ and 10¹¹ are 10 and 100 times 10⁹.
4. 20⁹ = 2⁹ × 10⁹ = 512 × 10⁹, far too big. (The book labels this first option (ii) by a misprint; it is option (i).)
5. So options (v) 2 × 10⁹ and (vi) 10⁹ + 10⁹ are both correct: they are the same number written two ways.

Answer2 × 10⁹: options (v) 2 × 10⁹ and (vi) 10⁹ + 10⁹

Watch this explained “Adding is not multiplying”, 3:28 into Exponential notation: repeated multiplication written once · हिंदी में देखें

Open NCERT p. 45Checked by computerAnswers can differ: one example

(i) If each person in the world had 30 pieces of clothing …

1. The question does not give the world's population. This chapter gives it as about 8.2 billion = 8.2 × 10⁹ (2025).
2. Total pieces = 8.2 × 10⁹ × 30 = 246 × 10⁹.
3. In scientific notation the number in front must be at least 1 and less than 10: 246 × 10⁹ = 2.46 × 10¹¹.

Answerabout 2.46 × 10¹¹ pieces of clothing

(ii) There are about 100 million bee colonies in the world

1. 100 million = 10⁸ colonies, and 50,000 = 5 × 10⁴ bees in each.
2. Total bees = 10⁸ × 5 × 10⁴ = 5 × 10^{8 + 4} = 5 × 10¹².

Answerabout 5 × 10¹² honeybees

(iii) The human body has about 38 trillion bacterial cells

1. 38 trillion = 38 × 10¹² = 3.8 × 10¹³ bacterial cells in one person.
2. People in the world ≈ 8.2 × 10⁹, as in (i).
3. Total = 3.8 × 10¹³ × 8.2 × 10⁹ = (3.8 × 8.2) × 10²² = 31.16 × 10²² = 3.116 × 10²³.

Answerabout 3.116 × 10²³ bacterial cells

(iv) Total time spent eating in a lifetime in seconds

1. Nothing is given, so assume, and say what you assumed: about 1 hour of eating a day, 365 days a year, over a 70-year life.
2. 1 hour = 60 × 60 = 3600 seconds.
3. Seconds spent eating = 3600 × 365 × 70 = 91,980,000.
4. In scientific notation: 9.198 × 10⁷ seconds.
5. Different assumptions give a different number, but for any sensible choice it stays around 10⁸ seconds.

Answerabout 9.198 × 10⁷ seconds (with 1 hour a day for 70 years)

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Open NCERT p. 45One way to think about it

1. 1 arab = 1 billion = 10⁹ seconds.
2. One day has 24 × 60 × 60 = 86,400 seconds.
3. 10⁹ ÷ 86,400 = 11,574 with 6,400 seconds left over, and 6,400 seconds = 1 hour 46 minutes 40 seconds. So 10⁹ seconds = 11,574 days, 1 hour, 46 minutes and 40 seconds.
4. In years: 11,574 ÷ 365.25 ≈ 31.69 years, which is about 31 years and 8 months (NCERT's answer also gives about 31.7 years).
5. To find the date, start from today's date and count back 31 years, then 8 months, then the few days that are left (about a week) to make 11,574 days in all.
6. The answer depends on the day you work it out, so there is no single date. Example: from 1 January 2026, going back 31 years and 8 months reaches 1 May 1994, which is 11,568 days earlier. Going back 6 more days to make 11,574 gives 25 April 1994.
7. The extra 1 hour 46 minutes 40 seconds changes the date by one more day only if the time now is before 1:46:40 a.m.

In shortAbout 31 years 8 months before today: 10⁹ seconds = 11,574 days (plus 1 hour 46 minutes 40 seconds). For example, from 1 January 2026 it was 25 April 1994.

Watch this explained “From a lifetime to the universe”, 8:11 into Why the nearest power of ten is the only handle on a quantity too big to picture · हिंदी में देखें