Powers and Roots (AQA GCSE Maths): Revision Notes

Powers and Roots

What are powers?

Powers provide a really handy way to write repeated multiplication in a much shorter form. Instead of writing out $2×2×2×2×2×2×2$, we can simply write $2^7$ (which we read as "two to the power of seven"). This makes calculations much easier to read and work with.

However, there are some important rules you need to master when working with powers. These rules will help you simplify expressions and solve problems efficiently.

The seven easy rules

Rule 1: Multiplying powers

When you're multiplying numbers that have the same base, you simply add their powers together. This works because you're essentially combining groups of the same repeated multiplication.

Mathematical rule: $a^m × a^n = a^{m+n}$

For example: $3^5 × 3^4 = 3^9$ and $a^6 × a^7 = a^{13}$

Rule 2: Dividing powers

When you're dividing numbers with the same base, you subtract the powers. This makes sense because division is the opposite of multiplication.

Mathematical rule: $a^m ÷ a^n = a^{m-n}$

For example: $5^8 ÷ 5^3 = 5^5$ and $b^9 ÷ b^5 = b^4$

Rule 3: Raising one power to another

When you have a power that's being raised to another power, you multiply the powers together. This is because you're repeating the repeated multiplication.

Mathematical rule: $(a^m)^n = a^{mn}$

For example: $(3^4)^2 = 3^8$ and $(c^3)^6 = c^{18}$

Rule 4: Anything to the power 1

Any number or variable raised to the power of 1 equals itself. This is because you're only multiplying the number by itself once.

Mathematical rule: $a^1 = a$

For example: $3^1 = 3$ and $d^1 = d$

Rule 5: Anything to the power 0

Any number or variable (except zero) raised to the power of 0 equals 1. This might seem strange at first, but it's a fundamental rule that makes other calculations work properly.

Mathematical rule: $a^0 = 1$ (where $a ≠ 0$)

For example: $5^0 = 1$ and $67^0 = 1$

Rule 6: 1 to any power

The number 1 raised to any power always equals 1. This is because multiplying 1 by itself any number of times will always give you 1.

Mathematical rule: $1^n = 1$

For example: $1^3 = 1$ and $1^{100} = 1$

Rule 7: Fractions with powers

When you have a fraction raised to a power, you apply that power to both the numerator (top) and denominator (bottom) separately.

Mathematical rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

For example: $\left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25}$ and $\left(\frac{y}{z}\right)^5 = \frac{y^5}{z^5}$

The three tricky rules

Rule 8: Negative powers

When you see a negative power, you need to "turn it upside-down" and make the power positive. This means the number becomes the reciprocal (1 divided by the number).

Mathematical rule: $a^{-n} = \frac{1}{a^n}$

For example: $7^{-2} = \frac{1}{7^2} = \frac{1}{49}$ and $a^{-1} = \frac{1}{a}$

Rule 9: Fractional powers

Fractional powers represent roots. The denominator of the fraction tells you what type of root it is.

The power $\frac{1}{2}$ means square root, the power $\frac{1}{3}$ means cube root, and the power $\frac{1}{4}$ means fourth root, and so on.

Mathematical rule: $a^{\frac{1}{n}} = \sqrt\[n]{a}$

For example: $25^{\frac{1}{2}} = \sqrt{25} = 5$ and $64^{\frac{1}{3}} = \sqrt\[3]{64} = 4$

Rule 10: Two-stage fractional powers

When you have more complex fractional powers like $64^{\frac{2}{3}}$, you need to tackle this in two stages. Split the fraction into a root and a power, then work through them in order: root first, then power.

Mathematical rule: $a^{\frac{m}{n}} = (a^{\frac{1}{n}})^m = (\sqrt\[n]{a})^m$

For example: $64^{\frac{2}{3}} = (64^{\frac{1}{3}})^2 = 4^2 = 16$