Year 9: Indices
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1. What are indices?
Indices (also called exponents or powers) are a shorthand way of expressing repeated multiplication. For example, 24 means 2 × 2 × 2 × 2 = 16.
In the expression an, a is the base and n is the index (or exponent).
2. Index notation examples
- 32 = 3 × 3 = 9
- 53 = 5 × 5 × 5 = 125
- 101 = 10
- 40 = 1 (any non-zero number to the power of 0 is 1)
3. Laws of indices
- Multiplying with the same base: am × an = am+n
- Dividing with the same base: am ÷ an = am−n
- Power of a power: (am)n = amn
- Zero index: a0 = 1 (a ≠ 0)
- Negative index: a−n = 1 ÷ an
4. Examples
- 23 × 22 = 25 = 32
- 56 ÷ 52 = 54 = 625
- (32)3 = 36 = 729
- 70 = 1
- 10−2 = 1 ÷ 102 = 1 ÷ 100 = 0.01
5. Scientific notation and indices
Scientific notation often uses powers of 10. For example:
- 1,000 = 1 × 103
- 0.001 = 1 × 10−3
The rules of indices apply when multiplying or dividing numbers in scientific notation.
6. Practice questions
- Evaluate 43
- Simplify: 35 × 32
- Simplify: 64 ÷ 61
- Evaluate: (23)2
- Write 1 ÷ 53 using a negative index
7. Solutions
- 4 × 4 × 4 = 64
- 37 = 2,187
- 63 = 216
- 26 = 64
- 5−3