Chapter focus
Powers are a compact way to describe repeated multiplication. They help us compare quantities that grow or shrink rapidly and reveal patterns in multiplication and division.
These original notes support revision and practice. Use your assigned NCERT edition for its complete examples, activities and exercise sequence.
1. Read and write powers
In 54, the number 5 is the base and 4 is the exponent. It means multiply four factors of 5: 5 × 5 × 5 × 5 = 625. The exponent tells how many times the base is used as a factor; it does not mean multiply 5 by 4.
For a non-zero number a, a1 = a. A zero exponent follows from the division rule: am ÷ am = 1, while subtracting exponents gives am−m = a0. Hence a0 = 1 for a ≠ 0. The expression 00 is not defined by this rule.
2. Laws of exponents
am × an = am+n. Add exponents because all factors remain.
am ÷ an = am−n, when a ≠ 0. Cancel common factors.
(am)n = amn. There are n groups of m factors.
(ab)n = anbn. The exponent applies to each factor.
Do not add exponents when the bases differ: 23 × 33 = (2 × 3)3 = 63, but 23 × 23 = 26.
3. Negative exponents and place value
For a non-zero base, a negative exponent represents a reciprocal: a−n = 1/an. This follows by extending the division law: 23 ÷ 25 = 2−2 = 1/22 = 1/4.
Powers of 10 make place value visible: 103 = 1,000, 100 = 1 and 10−2 = 1/100 = 0.01. Each step down by one exponent divides by 10. For example, 4.7 × 103 = 4,700, while 4.7 × 10−3 = 0.0047.
Scientific notation writes a number as c × 10n, where 1 ≤ c < 10 and n is an integer. Thus 58,000 = 5.8 × 104. This is useful for very large or very small measurements.
Worked examples
A. Simplify 35 ÷ 32
- Same base: subtract the exponents.
- 35−2 = 33.
- 3 × 3 × 3 = 27. Check by cancelling two factors in 243 ÷ 9.
B. Evaluate (23)2
- Use the power-of-a-power rule: 23×2 = 26.
- 26 = 64. Direct check: 8 × 8 = 64.
C. Write 0.00056 in scientific notation
Move the decimal four places right to make 5.6. The original number is smaller, so the exponent is negative: 0.00056 = 5.6 × 10−4.
Common errors to avoid
- (a + b)2 is not a2 + b2; expand the product if needed.
- am + an cannot usually be combined using an exponent law. The laws above apply to multiplication, division and powers.
- A negative exponent does not make the value negative: 2−3 = 1/8.
- For a negative base, use brackets carefully: (−2)4 = 16 but −24 = −16.
Quick revision
Same base ×: add exponents. Same non-zero base ÷: subtract exponents. A power raised to a power: multiply exponents. A negative exponent means reciprocal. Check with expanded multiplication when uncertain.
Practice: apply the rules
1. Simplify 73 × 72.
Answer: 75 = 16,807.
2. Evaluate (32)3 and explain the exponent.
Answer: 36 = 729. The three groups each contain two factors of 3, giving six factors altogether.
3. Write 1/100,000 as a power of 10.
Answer: 10−5, because 100,000 = 105.
4. A culture starts with 3 cells and triples every hour. How many cells are there after 4 hours?
Answer: 3 × 34 = 35 = 243 cells. The initial 3 cells are present at hour 0; each of four hours adds one factor of 3.
5. Which is greater: 210 or 103? Show a quick check.
Answer: 210 = 1,024 and 103 = 1,000, so 210 is greater.
Chapter summary
Exponents record repeated factors and make patterns in multiplication easy to see. Apply each law only when its base and operation match the rule, and use expanded multiplication or place value to check results.