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CBSE · Class 8 · Mathematics Part 1

Part 1, Chapter 2: Power Play

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Exponents and exponential form

From daily life

A paper folded once is 2 layers thick, folded twice is 4 layers, thrice is 8 layers. The layers keep doubling: 2, 2 × 2, 2 × 2 × 2, ... Writing such long products again and again is tiring, so we use powers.

Simple definition

nᵃ means n multiplied by itself a times. n is the base and a is the exponent (power). Example: 2 × 2 × 2 × 2 × 2 = 2⁵ = 32.

2¹22²42³82⁴162⁵322⁶64

In the figure: Each bar is double the one before it: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64.

Rule 1: Read and write powers — The exponent tells HOW MANY TIMES the base is multiplied, not what it is multiplied by.

  • Easy: 3⁵ = 3 × 3 × 3 × 3 × 3 = 243.
  • Medium: 2 × 3³ = 2 × 27 = 54 (the power acts only on 3).
  • Exam-type: 1800 = 2³ × 3² × 5² (divide by 2 three times, by 3 twice, by 5 twice).

Rule 2: Powers with letters — Count each letter-number separately and write its exponent.

  • Easy: y × y × y = y³.
  • Medium: 3 × 3 × k × k × k × k = 3²k⁴.
  • Exam-type: 4 + 4 + 4 = 3 × 4 = 12, but 4 × 4 × 4 = 4³ = 64.

Rule 3: Signs of powers — A negative base with an ODD exponent gives a negative answer; with an EVEN exponent a positive answer.

  • Easy: (−1)⁹ = −1.
  • Medium: (−1)⁴⁰ = +1.
  • Exam-type: (−3)³ × (−2)² = (−27) × 4 = −108.
Remember

Any number to the power 1 is itself (7¹ = 7) and 0 to a counting-number power is 0. Doubling is powerful: 2¹⁰ = 1024 already.

Common mistake

Confusing 5³ with 5 × 3 = 15. Correct: 5³ = 5 × 5 × 5 = 125.

Shortcut

To check the sign of a negative power, count the minus signs: odd → minus, even → plus. Memory clue: "Even is Positive, Odd is Negative".

Spot the difference

PointRepeated additionRepeated multiplication
Written as5 × 44⁵
Meaning4 + 4 + 4 + 4 + 44 × 4 × 4 × 4 × 4
Value201024

Practice questions

Easy
  1. 1Write 4 × 4 × 4 × 4 × 4 in exponential form.
  2. 2Find 2⁶.
  3. 3Find (−5)².
Medium
  1. 4Write 1800 as a product of prime powers.
  2. 5Find 2 × 3³.
  3. 6Find (−2)⁷ and say whether it is positive or negative.
Challenging
  1. 7A paper 0.004 cm thick is folded 6 times (doubling each time). Find its thickness.
  2. 8Which is greater, 3⁴ or 4³, and by how much?
Show answers (try them yourself first)
  1. 14⁵
  2. 264
  3. 325
  4. 42³ × 3² × 5²
  5. 554
  6. 6−128 (negative: odd power)
  7. 70.004 × 64 = 0.256 cm
  8. 83⁴ = 81 is greater than 4³ = 64 by 17

Multiplying powers and power of a power

From daily life

A sweet-seller has 3 trays; each tray has 3 boxes; each box has 3 laddoos. The laddoos are 3 × 3 × 3 = 3³. If every laddoo is packed again 3 times, we just multiply by 3 more times: 3³ × 3¹ = 3⁴. We never recount; we add the exponents.

Simple definition

Same base: nᵃ × nᵇ = nᵃ⁺ᵇ. Power of a power: (nᵃ)ᵇ = nᵃˣᵇ. Same exponent: nᵃ × mᵃ = (n × m)ᵃ.

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Multiplying powers and power of a power

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Rule 1: Same base: add the exponents — Write the base once and add the exponents. Do not multiply the bases.

  • Easy: 4⁵ × 4³ = 4⁸.
  • Medium: 2⁴ × 2 × 2³ = 2⁴⁺¹⁺³ = 2⁸ = 256.
  • Exam-type: (−2)³ × (−2)⁴ = (−2)⁷ = −128.

Rule 2: Power of a power: multiply the exponents — (nᵃ)ᵇ means nᵃ is multiplied b times, so there are a × b copies of n.

  • Easy: (2³)² = 2⁶ = 64.
  • Medium: (10²)³ = 10⁶ = 10,00,000.
  • Exam-type: 9² × 3⁴: write 9 = 3², so 9² = 3⁴ and 3⁴ × 3⁴ = 3⁸ = 6561.

Rule 3: Same exponent: multiply the bases — If two powers have the same exponent, put the bases together under one exponent.

  • Easy: 2³ × 5³ = 10³ = 1000.
  • Medium: 2⁵ × 3⁵ = 6⁵ = 7776.
  • Exam-type: 16⁵ × 4³: 16⁵ = (2⁴)⁵ = 2²⁰ and 4³ = 2⁶, so the product is 2²⁶.
Remember

Only powers with the same base can be added in the exponent: 2³ × 3² is NOT 6⁵. And (nᵃ)ᵇ = (nᵇ)ᵃ because a × b = b × a.

Common mistake

Multiplying the bases as well: 6² × 6² is 6⁴, not 36⁴ or 12⁴. Also mixing the rules: 5³ × 5² = 5⁵ (add) but (5³)² = 5⁶ (multiply).

Shortcut

"Multiply gives ADD; power-of-power gives MULTIPLY" - the operation changes. Memory clue: "× → +, ( )ᵃ → ×".

Spot the difference

Pointnᵃ × nᵇ(nᵃ)ᵇ
Ruleadd exponents: nᵃ⁺ᵇmultiply exponents: nᵃˣᵇ
Example3² × 3⁴ = 3⁶(3²)⁴ = 3⁸
Value7296561

Practice questions

Easy
  1. 1Simplify 4⁵ × 4³.
  2. 2Find (2³)².
  3. 3Simplify y⁷ × y².
Medium
  1. 4Find 2⁵ × 3⁵.
  2. 5Write (10²)³ with a single exponent.
  3. 6Find 3⁴ × 3⁴ and write it as one power of 3.
Challenging
  1. 7Write 16⁵ × 4³ as a power of 2.
  2. 8Find k if 7ᵏ × 7⁴ = 7¹¹.
Show answers (try them yourself first)
  1. 14⁸
  2. 264
  3. 3y⁹
  4. 47776
  5. 510⁶
  6. 63⁸ = 6561
  7. 72²⁶
  8. 87

Dividing powers, zero and negative exponents

From daily life

A shopkeeper packs 2⁸ pencils and sells 2⁵ of them. The pencils left are 2⁸ ÷ 2⁵ = 2³ = 8 (subtract the exponents). If he sells all 2⁸, what is left is 2⁸ ÷ 2⁸ = 2⁰ = 1 as a ratio.

Simple definition

nᵃ ÷ nᵇ = nᵃ⁻ᵇ (n ≠ 0). So n⁰ = 1 and n⁻ᵃ = 1/nᵃ. Also nᵃ ÷ mᵃ = (n ÷ m)ᵃ.

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Dividing powers, zero and negative exponents

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Rule 1: Same base: subtract the exponents — Write the base once and subtract the lower exponent from the upper one.

  • Easy: 6⁹ ÷ 6⁴ = 6⁵.
  • Medium: 10⁶ ÷ 10² = 10⁴ = 10,000.
  • Exam-type: (3⁴ × 3⁻¹) ÷ 3² = 3³ ÷ 3² = 3.

Rule 2: Zero power — Any non-zero number to the power 0 is 1, because nᵃ ÷ nᵃ = 1 and also = n⁰.

  • Easy: 8⁰ = 1.
  • Medium: 5⁰ + 3⁰ = 1 + 1 = 2.
  • Exam-type: (−7)⁰ = 1 and 100⁰ = 1.

Rule 3: Negative power = reciprocal — n⁻ᵃ is 1 divided by nᵃ. A negative exponent flips a fraction: (a/b)⁻ᵐ = (b/a)ᵐ.

  • Easy: 2⁻⁴ = 1/2⁴ = 1/16.
  • Medium: 10⁻⁵ = 1/10⁵ = 0.00001.
  • Exam-type: (3/5)⁻³ = (5/3)³ = 125/27.
Remember

A negative exponent does NOT make the number negative: 2⁻³ = 1/8, which is positive. And 0⁰ is not defined here; only n ≠ 0 has n⁰ = 1.

Common mistake

Writing 2⁻³ = −8 or −6. Correct: 2⁻³ = 1/2³ = 1/8. Another slip: 6⁵ ÷ 6⁷ = 6⁻² = 1/36, not 6².

Shortcut

"Minus below, flip over": a minus sign in the exponent means flip the base to the bottom. Memory clue: n⁻ᵃ = "1 over nᵃ".

Spot the difference

PointPositive exponentNegative exponent
Meaningnᵃ = n × n × … (a times)n⁻ᵃ = 1 ÷ nᵃ
Example2³ = 82⁻³ = 1/8
Is the value bigger than 1?yes (when n > 1)no, it is a fraction

Practice questions

Easy
  1. 1Simplify 6⁹ ÷ 6⁴.
  2. 2Find 8⁰.
  3. 3Find 2⁻⁴.
Medium
  1. 4Find 15³ ÷ 5³.
  2. 5Write 10⁻⁵ as a decimal.
  3. 6Find 3⁻¹ + 6⁻¹.
Challenging
  1. 7Simplify (4⁻² × 4⁵) ÷ 4².
  2. 8Find (3/5)⁻³.
Show answers (try them yourself first)
  1. 16⁵
  2. 21
  3. 31/16
  4. 427
  5. 50.00001
  6. 61/2
  7. 74 (4⁻²⁺⁵⁻² = 4¹)
  8. 8125/27

Powers of 10 and scientific notation

From daily life

The distance from the Earth to the Sun is about 150000000 km. Writing and comparing so many zeros is hard and we can miscount them. Scientists write it as 1.5 × 10⁸ km.

Simple definition

Scientific notation (standard form) writes a number as x × 10ʸ where 1 ≤ x < 10 and y is an integer. 10ʸ for positive y has y zeros; 10⁻ʸ = 1/10ʸ.

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Powers of 10 and scientific notation

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Rule 1: Big numbers: move the decimal point left — Move the point until one non-zero digit is before it. The number of steps is the positive power of 10.

  • Easy: 800 = 8 × 10².
  • Medium: 3,60,000 = 3.6 × 10⁵.
  • Exam-type: 25,00,000 = 2.5 × 10⁶ (move 6 steps).

Rule 2: Small numbers: move the decimal point right — Move the point right until one non-zero digit is before it. The steps give a NEGATIVE power.

  • Easy: 0.03 = 3 × 10⁻².
  • Medium: 0.0004 = 4 × 10⁻⁴.
  • Exam-type: 0.00000062 = 6.2 × 10⁻⁷.

Rule 3: Multiply and divide in this notation — Multiply or divide the numbers, and add or subtract the powers of 10.

  • Easy: (2 × 10³) × (3 × 10⁴) = 6 × 10⁷.
  • Medium: (4 × 10⁵) × (2 × 10³) = 8 × 10⁸.
  • Exam-type: (9 × 10⁷) ÷ (3 × 10⁴) = 3 × 10³.
Remember

In x × 10ʸ the number x is at least 1 and less than 10. So 0.42 × 10⁶ is NOT in standard form; it is 4.2 × 10⁵.

Common mistake

Moving the point the wrong way: 45000 = 4.5 × 10⁴ (not 4.5 × 10⁻⁴). Big numbers get a positive power; numbers below 1 get a negative power.

Shortcut

Count the zeros of a big number: that count is the power when the number is 1 followed by zeros (10⁷ = 1 crore). Memory clue: "Big → Positive, Small → Negative power".

Spot the difference

PointBig numberSmall number
Example5,000 = 5 × 10³0.005 = 5 × 10⁻³
Decimal point movesleftright
Power of 10positivenegative

Practice questions

Easy
  1. 1Write 10⁴ as an ordinary number.
  2. 2Write 800 in scientific notation.
  3. 3Write 2.5 × 10³ as an ordinary number.
Medium
  1. 4Write 3,60,000 in scientific notation.
  2. 5Write 0.0004 in scientific notation.
  3. 6Find (4 × 10⁵) × (2 × 10³).
Challenging
  1. 7Find (9 × 10⁷) ÷ (3 × 10⁴).
  2. 8A train goes 1 × 10² km every hour. How many hours does it need to cover 1.5 × 10⁸ km?
Show answers (try them yourself first)
  1. 110,000
  2. 28 × 10²
  3. 32500
  4. 43.6 × 10⁵
  5. 54 × 10⁻⁴
  6. 68 × 10⁸
  7. 73 × 10³
  8. 81.5 × 10⁶ hours

Large numbers and comparing powers

From daily life

Plan A: your pocket money is ₹2 on day 1 and doubles every day. Plan B: ₹50 on day 1, ₹100 on day 2, and so on. At first B is much more, but A catches up and flies ahead. That is the power of multiplicative growth.

Simple definition

Additive growth adds the same amount every step (5n). Multiplicative (exponential) growth multiplies by the same factor every step (2ⁿ). To compare powers, make the bases or the exponents the same, or use 2¹⁰ ≈ 10³.

123456doubling (2ⁿ)adding 5 (5n)day

In the figure: Orange (adding 5 each day) leads for the first days, but purple (doubling) overtakes it and keeps growing faster.

Rule 1: Compare with the same base or exponent — Rewrite both numbers so that either the base or the exponent matches, then compare the other part.

  • Easy: 2⁷ = 128 is greater than 7² = 49.
  • Medium: 4¹⁰ = (2²)¹⁰ = 2²⁰, so 4¹⁰ = 2²⁰.
  • Exam-type: 3⁴⁰ = (3²)²⁰ = 9²⁰ and 2⁶⁰ = (2³)²⁰ = 8²⁰; since 9 > 8, 3⁴⁰ is greater.

Rule 2: Estimate with 2¹⁰ ≈ 10³ — 2¹⁰ = 1024 is very close to 1000, so every 10 doublings is about 3 zeros.

  • Easy: 2²⁰ ≈ 10⁶ (about a million).
  • Medium: 2³⁰ ≈ 10⁹ (about a billion).
  • Exam-type: 2⁴⁰ = (2¹⁰)⁴ ≈ (10³)⁴ = 10¹².

Rule 3: Count digits and times bigger — 10ⁿ has n + 1 digits. aᵐ ÷ aⁿ = aᵐ⁻ⁿ tells how many times bigger.

  • Easy: 10¹⁰ has 11 digits.
  • Medium: 10⁴ is 10⁴ ÷ 10² = 100 times 10².
  • Exam-type: 2¹² = 4096, so doubling a quantity 12 times multiplies it by 4096.
Remember

Doubling 10 times makes a quantity about 1000 times bigger; 20 doublings about a million times. A small base with a big exponent can beat a big base with a small exponent.

Common mistake

Judging by the base only: 2⁷ = 128 is larger than 7² = 49 even though 7 > 2. Always compute or rewrite both.

Shortcut

Rewrite with the same exponent: to compare 3⁴⁰ and 2⁶⁰, take out the common factor 20 in the exponent. Memory clue: "Same exponent? compare bases. Same base? compare exponents."

Spot the difference

PointAdditive growthMultiplicative growth
Each stepadd the same numbermultiply by the same number
Example (start 1)1, 6, 11, 16 …1, 2, 4, 8 …
Speedsteadybecomes huge very fast

Practice questions

Easy
  1. 1Which is greater, 2⁷ or 7²?
  2. 2How many times is 10⁴ bigger than 10²?
  3. 3How many digits does 10¹⁰ have?
Medium
  1. 4A cell count of 1 doubles 12 times. How many cells are there?
  2. 5Using 2¹⁰ ≈ 10³, estimate 2³⁰.
  3. 6Compare 4¹⁰ and 2²⁰.
Challenging
  1. 7Which is greater, 3⁴⁰ or 2⁶⁰?
  2. 8Plan A: ₹2 on day 1, doubling daily (2, 4, 8, …). Plan B: ₹50 × day number (50, 100, 150, …). On which day does Plan A first exceed Plan B?
Show answers (try them yourself first)
  1. 12⁷ = 128 (7² = 49)
  2. 2100
  3. 311
  4. 44096
  5. 5≈ 10⁹
  6. 6Equal: 4¹⁰ = (2²)¹⁰ = 2²⁰
  7. 73⁴⁰ (9²⁰ > 8²⁰)
  8. 8Day 9 (512 > 450); on day 8 it is 256 < 400