Powers, Roots and Standard Form

Number

Law	What it does	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	Multiplying adds the indices	5⁶ × 5³ = 5⁹
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	Dividing subtracts the indices	9⁷ ÷ 9⁴ = 9³
(aᵐ)ⁿ = aᵐⁿ	A power of a power multiplies the indices	(2⁵)³ = 2¹⁵
(ab)ⁿ = aⁿbⁿ	A power spreads across a product	(4 × 7)² = 4² × 7²
a⁰ = 1	Any non-zero base to the power 0	12⁰ = 1
a⁻ⁿ = 1 ÷ aⁿ	A negative index gives the reciprocal	3⁻² = 1/9
a^(1/n) = ⁿ√a	A unit fraction index is a root	16^(1/2) = 4
a^(m/n) = (ⁿ√a)ᵐ	Root first, then power	27^(2/3) = 3² = 9

Taking the root before applying the power keeps the numbers small, which matters without a calculator
A negative index never makes the answer negative: it makes it a reciprocal

Where the two bases differ but one is a power of the other, rewrite so that they match, and the laws then apply
- 4 = 2², so 2⁵ × 4³ = 2⁵ × (2²)³ = 2⁵ × 2⁶ = 2¹¹
Powers of 2, 3 and 5 are the ones worth recognising, since those are the bases questions use
Bases with no such relationship stay apart, so 2³ × 5² cannot be shortened by these laws at all

Worked example

Using the laws, and matching the bases

Simplify 2⁵ × 2³ ÷ 2², and solve 4ˣ = 8.

Solution:

Multiplying powers of the same base adds the indices: 2⁵ × 2³ = 2⁸
Dividing subtracts them: 2⁸ ÷ 2² = 2⁶
So the expression is 2⁶, which is 64
For the equation, the two sides have different bases, so rewrite both as powers of 2
4 = 2², so 4ˣ = (2²)ˣ = 2²ˣ, and 8 = 2³
The equation becomes 2²ˣ = 2³
Equal powers of the same base must have equal indices, so 2x = 3 and x = 1.5