Power and Roots

Index notation is a short way to write a number multiplied by itself many times. Instead of $3 \times 3 \times 3 \times 3$, you write $3^4$.

In $3^4$, the large number is the base and the small raised number is the index (also called the power or exponent). The index counts how many times the base appears.

So $3^4 = 3 \times 3 \times 3 \times 3 = 81$. The index 2 has the name "squared" and the index 3 the name "cubed"; every other index is read "to the power of".

Index notation fits inside BIDMAS — the I stands for indices, so a power is worked out after brackets but before any multiplication, division, addition or subtraction. In $5 + 2^3$ you do $2^3 = 8$ first, then $5 + 8 = 13$. Getting comfortable with the base/index language is what makes the laws of indices, coming next, easy to follow.

Here is the first law of indices. To multiply two powers that share a base, you simply add the indices.

Why does this work? Write it out in full. $2^3 \times 2^2$ means $(2 \times 2 \times 2) \times (2 \times 2)$ — that is five 2s multiplied, which is $2^5$. The indices 3 and 2 added together give 5.

The rule in symbols is $a^m \times a^n = a^{m+n}$. So $5^4 \times 5^3 = 5^7$, and $x^2 \times x^6 = x^8$.

There is one condition that matters: the base must be the same. The law does not let you simplify $2^3 \times 5^2$, because 2 and 5 are different bases — there is nothing to combine.

The second law mirrors the first. To divide two powers that share a base, you subtract the indices.

Again, writing it in full shows why. $2^5 \div 2^2$ is $\frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2}$. Two of the 2s on top cancel with the two on the bottom, leaving three 2s — that is $2^3$. The indices 5 and 2 subtract to give 3.

The rule in symbols is $a^m \div a^n = a^{m-n}$. So $7^8 \div 7^3 = 7^5$, and $x^9 \div x^4 = x^5$.

This law also explains a neat fact. What is $a^0$? Think of $a^3 \div a^3$: by the law that is $a^{3-3} = a^0$, but a number divided by itself is 1. So any non-zero number to the power 0 equals 1 — for example $6^0 = 1$.

The third law deals with a power inside brackets, raised to another power. Here you multiply the indices.

Take $(2^3)^2$. The outer index 2 says "use $2^3$ twice": $(2^3)^2 = 2^3 \times 2^3$. By the multiplying law, that is $2^{3+3} = 2^6$. Multiplying the indices, $3 \times 2 = 6$, reaches the same answer faster.

The rule in symbols is $(a^m)^n = a^{mn}$. So $(5^2)^4 = 5^8$, and $(x^3)^3 = x^9$.

It is worth keeping the three laws clearly apart: multiply powers → add indices; divide powers → subtract indices; power of a power → multiply indices. Saying which operation you are doing first stops the laws getting tangled.

Squaring and cubing have opposites that undo them — roots. The square root of a number asks "what was squared to get this?" — since $6^2 = 36$, $\sqrt{36} = 6$. The cube root asks "what was cubed?" — since $4^3 = 64$, $\sqrt[3]{64} = 4$.

The fastest way to find a root is to recognise it from the square and cube numbers you know. So learning the squares to $15^2 = 225$ and the cubes to $5^3 = 125$ really pays off.

Powers of 10 deserve special attention because our number system is built on them.

The index equals the number of zeros: $10^3 = 1000$. Place value columns are themselves powers of 10. Rather than writing two million as $2\,000\,000$, you can describe it as $2 \times 10^6$ — shorthand you will meet again as "standard form".

The skills in this guide open up a lot of later maths:

• Algebra uses the laws of indices constantly — simplifying $x^3 \times x^4$ to $x^7$ is the very same rule you used with numbers.
• Standard form in later stages relies entirely on powers of 10 to handle very large and very small numbers.
• Area and volume are full of squares and cubes — a square's area is a side squared, a cube's volume is an edge cubed.
• Prime factorisation is written far more neatly using index notation, such as $360 = 2^3 \times 3^2 \times 5$.

If a later topic involving powers feels tricky, it is usually worth a quick refresh of the three index laws and the square and cube number lists.