Power and Roots

A square number is the answer when you multiply a whole number by itself. $5 \times 5 = 25$, so 25 is a square number. We write this as $5^2$, said "five squared".

The name "square" is not random. If you arrange 25 dots into a grid, they make a perfect square that is 5 dots wide and 5 dots tall.

The first few square numbers are $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$. It is well worth learning them up to $15^2 = 225$ by heart, because they pop up constantly in area, in mental arithmetic, and in later topics.

Notice the gaps between square numbers grow by the odd numbers: $1, 3, 5, 7, 9 \dots$ That neat pattern is a quick way to check you have the list right.

A cube number is the answer when a whole number is used three times in a multiplication. $4 \times 4 \times 4 = 64$, so 64 is a cube number, written $4^3$ and said "four cubed".

Just like squares form a flat square, cube numbers form a solid cube. $3^3 = 27$ small blocks stack into a cube that is 3 blocks wide, 3 deep and 3 tall.

The first few cube numbers are $1, 8, 27, 64, 125, 216, 1000$. They grow much faster than square numbers, so the list gets large quickly. Knowing the cubes up to $5^3 = 125$ and the handy fact $10^3 = 1000$ covers nearly everything you will need at this stage.

Squaring and cubing have opposites that undo them, called roots.

The square root of a number asks: "what was squared to get this?" Since $6^2 = 36$, the square root of 36 is 6, written $\sqrt{36} = 6$.

The cube root asks: "what was cubed to get this?" Since $4^3 = 64$, the cube root of 64 is 4, written $\sqrt[3]{64} = 4$.

The easiest way to find a root is to use the square and cube numbers you already know. To find $\sqrt{81}$, run through the squares until you reach 81 — that is $9^2$, so $\sqrt{81} = 9$.

Most numbers do not have a tidy whole-number root. At this stage you only need the roots of numbers that are perfect squares or cubes, which is exactly why learning those lists pays off.

Writing $2 \times 2 \times 2 \times 2 \times 2$ is slow and easy to miscount. Index notation packs it up neatly as $2^5$.

In $2^5$, the big number is the base and the small raised number is the index (also called the power or exponent). The index simply counts how many times the base appears in the multiplication.

So $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$. The index 2 has the special name "squared" and the index 3 the name "cubed", but every other index is just read as "to the power of" — $2^5$ is "two to the power of five".

Index notation also fits inside the order of operations. In BIDMAS the I stands for indices, so a power is worked out before any multiplication, division, addition or subtraction (but after brackets). In $3 + 2^3$ you do $2^3 = 8$ first, then $3 + 8 = 11$.

Powers of 10 deserve their own spotlight because our whole number system is built on them.

Look at the pattern: $10^1 = 10$, $10^2 = 100$, $10^3 = 1000$, $10^4 = 10\,000$. The index tells you how many zeros follow the 1.

This is exactly how place value works. The digits of a number stand for ones, tens, hundreds and thousands — and those are $10^0, 10^1, 10^2, 10^3$. When you multiply a number by 10 you slide every digit one place to the left, which is the same as adding a zero.

Powers of 10 also make huge numbers manageable. Rather than writing two million as $2\,000\,000$, you can describe it as $2 \times 10^6$. You will meet this shorthand again as "standard form" in later stages.

Getting comfortable with powers and roots now opens up a lot of topics ahead:

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

If a later topic involving squares or roots feels tricky, it is usually worth a quick refresh of the square and cube number lists. Knowing those by heart is one of the best time-savers in all of maths.