Square roots explained: perfect squares, surds and estimating

The square root of a number is the value that, multiplied by itself, gives that number: $\sqrt{49} = 7$ because $7 \times 7 = 49$. Every positive number has two square roots, $7$ and $-7$, but the symbol $\sqrt{\;}$ always means the positive one.

Perfect squares to know

Knowing the first squares makes every root problem faster: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Their square roots are whole numbers; the square root of any other whole number is irrational — an endless, non-repeating decimal.

Simplifying square roots (surds)

To simplify, split the number into a perfect square times something else and take the root of the square: $\sqrt{ab} = \sqrt{a}\sqrt{b}$ for numbers that aren’t negative.

Estimating a square root by hand

For a number that isn’t a perfect square, find the two perfect squares it sits between. $65$ is between $64 = 8^2$ and $81 = 9^2$, and much closer to 64, so $\sqrt{65}$ is just over 8.

Cube roots

A cube root undoes cubing: $\sqrt[3]{27} = 3$ because $3^3 = 27$. Cube roots of negative numbers exist: $\sqrt[3]{-8} = -2$. Simplify them by taking out cube factors: $\sqrt[3]{54} = \sqrt[3]{27 \times 2} = 3\sqrt[3]{2}$.

Common mistakes

• Adding under the root: $\sqrt{9 + 16} = 5$, not $3 + 4 = 7$.
• Forgetting the negative solution of $x^2 = 25$: $x = \pm 5$.
• Not using the largest square factor: $\sqrt{180} = 3\sqrt{20}$ isn’t fully simplified.
• Rounding too early in a longer calculation.