Powers and roots

Squares and square roots

Squaring a number multiplies it by itself: $7^2=7\times 7=49$. The square root undoes squaring: $\sqrt{49}=7$, because $7^2=49$.

Every positive number has two square roots, one positive and one negative, because $(-7{)}^2=49$ as well. The root sign on its own always means the positive root, so $\sqrt{49}=7$. When you solve $x^2=49$, you need both answers: $x=7$ or $x=-7$, written $x=\pm 7$.

A negative number has no square root among the numbers used at IGCSE, because squaring a number never gives a negative answer. $\sqrt{-9}$ does not exist.

You must know the squares from $1^2$ to ${15}^2$, and so their square roots, without a calculator. Reading the table backwards gives the roots: $\sqrt{169}=13$ and $\sqrt{225}=15$.

Squares to learn

Notice the pattern in the last digits: a square number never ends in 2, 3, 7 or 8. That is a quick way to see that 147 cannot be a square.

Cubes and cube roots

Cubing uses a number three times in a multiplication: $4^3=4\times 4\times 4=64$. The cube root undoes it: $\sqrt[3]{64}=4$, because $4^3=64$.

Learn the cubes of 1, 2, 3, 4, 5 and 10, and their cube roots.

Unlike square roots, cube roots of negative numbers do exist. $(-2{)}^3=(-2)\times (-2)\times (-2)=-8$, so $\sqrt[3]{-8}=-2$. A cube root keeps the sign of the number, and every number has exactly one cube root.

Cubes to learn

So $\sqrt[3]{1}=1$, $\sqrt[3]{8}=2$, $\sqrt[3]{27}=3$, $\sqrt[3]{64}=4$, $\sqrt[3]{125}=5$ and $\sqrt[3]{1000}=10$.

Other powers and roots

Higher powers work in the same way: $2^5=2\times 2\times 2\times 2\times 2=32$ and $3^4=3\times 3\times 3\times 3=81$. The matching roots are written with a small number on the root sign: $\sqrt[5]{32}=2$ and $\sqrt[4]{81}=3$.

The powers of 2 are worth knowing: $2,4,8,16,32,64,128,256,512,1024$. Powers of 10 count zeros: ${10}^6=1000000$.

Powers of negative numbers. Each pair of negatives multiplies to a positive, so an even power of a negative number is positive and an odd power is negative: $(-2{)}^4=16$ but $(-2{)}^5=-32$.

Brackets matter. A power applies only to what it is attached to. $(-3{)}^2=9$, but $-3^2$ means $-(3^2)=-9$.

Order of operations

Powers and roots are worked out before multiplication and division, which come before addition and subtraction. Brackets come first of all. So

$$5^2\times \sqrt[3]{8}=25\times 2=50$$

A root sign acts like a bracket around everything under it. $\sqrt{9+16}=\sqrt{25}=5$. It is not $\sqrt{9}+\sqrt{16}=3+4=7$: the root of a sum is not the sum of the roots.

Roots of decimals and fractions without a calculator

Fractions. Take the root of the top and of the bottom separately: $\sqrt{\frac{9}{25}}=\frac{3}{5}$ and $\sqrt[3]{\frac{8}{125}}=\frac{2}{5}$.

Decimals. Squaring doubles the number of decimal places, and cubing triples it:

• ${0.6}^2=0.36$, so $\sqrt{0.36}=0.6$;
• ${0.03}^2=0.0009$, so $\sqrt{0.0009}=0.03$;
• ${0.2}^3=0.008$, so $\sqrt[3]{0.008}=0.2$.

Why: $0.6=\frac{6}{10}$, so ${0.6}^2=\frac{36}{100}$. Squaring the 10 underneath makes 100, which is two decimal places.

Squaring a number between 0 and 1 makes it smaller: ${0.3}^2=0.09$, not 0.9. A good check is to ask whether your answer should be bigger or smaller than the number you started with.

Multiples of 10. ${70}^2=4900$, so $\sqrt{4900}=70$. Pairs of zeros come out of a square root as single zeros; groups of three zeros come out of a cube root as single zeros: $\sqrt[3]{8000}=20$.

Estimating roots

When a number is not a perfect square, place its root between two whole numbers using the squares you know. $49<50<64$, so $7<\sqrt{50}<8$, and since 50 is much nearer 49 than 64, $\sqrt{50}$ is just over 7. In the same way $\sqrt[3]{30}$ lies between 3 and 4, because $27<30<64$.

Estimating is a check on calculator work: if your calculator says $\sqrt{50}=2.5$ you have pressed the wrong key.

Using a calculator

On the calculator papers use the keys for $x^2$, $x^3$, square root, cube root and the power key (marked $x^y$ or ^) for any other power. Most calculators also have a key for other roots.

Two habits save marks:

1. Brackets. Put the whole of anything under a root, or on the top or bottom of a fraction, inside brackets, or use the fraction key. For $\frac{{2.4}^3-1.8}{\sqrt{5.2}}$, the top must be worked out before dividing.
2. Round at the end. Write down more figures than you need from the display, then round the final answer, usually to 3 significant figures. Rounding part way through can change the last figure.