Laws of indices · Year 12
In Year 12 you learn: solving equations with indices.
Index laws are shortcuts for multiplying and dividing powers of the same number. They follow straight from what a power means, and they extend powers to zero, negative and fractional indices, which are used throughout algebra, science and finance.
Key facts
- Multiply: add the indices. x^a × x^b = x^(a + b)
- Divide: subtract the indices. x^a ÷ x^b = x^(a − b)
- Power of a power: multiply the indices. (x^a)^b = x^(ab)
- Zero index: x^0 = 1 (for any x except 0).
- Negative index means 1 over: x^(−n) = 1/(x^n), so 2^(−3) = 1/8.
- Fractional index means a root: x^(1/n) = ⁿ√x and x^(m/n) = (ⁿ√x)^m. For example, 8^(2/3) = 2^2 = 4.
Worked examples
Example 1
Simplify (3a^4)^2.
Answer: 9a^8
Example 2
Solve 4^x = 32.
Answer: x = 5/2