Powers and Roots

When the same number is multiplied by itself several times, we write it using a power (also called an index or exponent). The lower number is the base, and the small raised number is the index — it counts how many times the base is being multiplied.

So $a^n$ means $a \times a \times a \times \dots \times a$ with $n$ copies of $a$. For example $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.

Two special names you will use often:

• $a^2$ is read 'a squared' (because the area of a square with side $a$ is $a \times a$).
• $a^3$ is read 'a cubed' (because the volume of a cube with side $a$ is $a \times a \times a$).

Anything to the power $1$ is just the number itself: $7^1 = 7$. And $a^0 = 1$ for any non-zero base — a result you will meet again with the division law of indices.

A square number is what you get when a whole number is squared. The first few are $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, \dots$. Knowing them all the way up to $15^2 = 225$ by heart pays for itself many times over.

A cube number is what you get when a whole number is cubed: $1, 8, 27, 64, 125, 216, \dots$. The cubes from $1^3$ to $5^3$ are the ones to memorise.

A root undoes a power. The square root $\sqrt{n}$ asks "what number, multiplied by itself, gives $n$?". The cube root $\sqrt[3]{n}$ asks "what number, cubed, gives $n$?". So $\sqrt{49} = 7$ because $7 \times 7 = 49$, and $\sqrt[3]{125} = 5$ because $5 \times 5 \times 5 = 125$.

For roots that are not whole numbers, you might estimate them. For example $\sqrt{50}$ sits between $\sqrt{49} = 7$ and $\sqrt{64} = 8$, and is just a touch above $7$. A calculator gives $\sqrt{50} \approx 7.07$.

Three rules cover almost every index calculation. They only work when the bases are the same.

• Multiplication rule: $a^m \times a^n = a^{m+n}$. Multiplying powers of the same base means adding the indices. So $2^3 \times 2^4 = 2^7 = 128$.
• Division rule: $a^m \div a^n = a^{m-n}$. Dividing means subtracting the indices. So $7^9 \div 7^4 = 7^5$.
• Power-of-a-power rule: $(a^m)^n = a^{mn}$. Raising a power to another power means multiplying the indices. So $(5^3)^2 = 5^6$.

The division rule also explains why $a^0 = 1$. Look at $a^3 \div a^3$: dividing anything by itself gives $1$, but the rule says it equals $a^{3-3} = a^0$. So $a^0$ must be $1$ for any non-zero $a$.

These three rules can be combined freely. Simplify $\dfrac{x^4 \times x^5}{x^3}$: numerator is $x^{4+5} = x^9$, then divide $x^9 \div x^3 = x^{6}$.

Raising a negative number to a power is just repeated multiplication, so the sign-counting rule from integer work applies. $(-3)^4$ means $-3 \times -3 \times -3 \times -3$ — four negatives, an even count, so the answer is positive $81$. But $(-3)^3$ has three negatives, an odd count, so it is negative $-27$.

The trap to meet head-on is the difference between $(-4)^2$ and $-4^2$:

• $(-4)^2$ means the whole of $-4$ is squared: $-4 \times -4 = 16$.
• $-4^2$ means only the 4 is squared, and the minus is applied afterwards: $-(4 \times 4) = -16$.

The brackets decide what the index acts on. Whenever you raise a negative number to a power, wrap it in brackets so there is no doubt at all.

Once you know prime factorisation, you have a powerful tool for roots of large numbers. If you can write a number as a product of pairs of primes, it is a square number. If you can group its primes into triples, it is a cube number.

For example, is $324$ a square number?

• Prime factorise: $324 = 2^2 \times 3^4 = (2 \times 3 \times 3) \times (2 \times 3 \times 3)$.
• Every prime appears an even number of times, so $324 = 18^2$ and $\sqrt{324} = 18$.

For a cube root, group the primes into triples instead. $\sqrt[3]{216}$: prime factorise to get $216 = 2^3 \times 3^3 = (2 \times 3) \times (2 \times 3) \times (2 \times 3)$. So $\sqrt[3]{216} = 6$.

If a number's prime factorisation does not pair up neatly (for $\sqrt{}$) or triple up (for $\sqrt[3]{}$), then its root is not a whole number — you'd use a calculator instead.

Confident work with powers and roots is essential for:

• Algebra — simplifying expressions like $x^4 \times x^3$ or $\dfrac{y^7}{y^2}$ relies on the index laws.
• Standard form — every number is written using a power of 10.
• Pythagoras' theorem — uses squares and square roots: $c = \sqrt{a^2 + b^2}$.
• Area and volume — area uses squares, volume uses cubes, in obvious ways.
• Science and finance — exponential growth and decay all rely on powers.

The index laws and the small squares and cubes are tools you will reach for again and again. Make them automatic now and the rest of secondary maths is much smoother.