Simplifying surds and powers of surds

A surd is in its sim­plest form when the num­ber under the root has no square fac­tor left in it (for a square root). $\sqrt{72}$ is not in sim­plest form; $6\sqrt{2}$ is, and it is the same num­ber. Sim­pli­fy­ing surds lets you add and sub­tract them, com­pare them, and spot answers quickly. This les­son also shows how to work out pow­ers of surds such as $(\sqrt{2})^{11}$ with­out a cal­cu­la­tor.

Sim­pli­fy­ing a square root#

Split the num­ber under the root into a per­fect square times another fac­tor, using the largest per­fect square you can. Then take the square root of the per­fect square out­side, by Law 1:

$$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$$

The per­fect squares to look for are $4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots$ Learn the results below; they come up con­stantly.

Use the largest square fac­tor. $\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}$ is true but unfin­ished, because $18$ still has the square fac­tor $9$. Using $36$ straight away gives $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.

From a mixed surd to a pure surd#

Going the other way, a mixed surd becomes a pure surd by squar­ing the out­side num­ber and mov­ing it under the root:

$$3\sqrt{2} = \sqrt{9} \times \sqrt{2} = \sqrt{18}$$

This is handy for com­par­ing surds. Which is big­ger, $3\sqrt{2}$ or $2\sqrt{3}$? As pure surds they are $\sqrt{18}$ and $\sqrt{12}$, so $3\sqrt{2}$ is big­ger.

Adding and sub­tract­ing surds#

Surds with the same surd part, such as $3\sqrt{2}$ and $5\sqrt{2}$, are called like surds. Like surds add and sub­tract just like like terms in alge­bra: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$, in the same way that $3x + 5x = 8x$. Unlike surds, such as $\sqrt{2}$ and $\sqrt{3}$, can­not be com­bined.

Surds that look unlike may become like once sim­pli­fied. So sim­plify first, then col­lect.

Exam­ple 1#

If $\sqrt{18} + \sqrt{72} + 5\sqrt{32} = k\sqrt{2}$, find $k$.

$$\begin{aligned} \sqrt{18} + \sqrt{72} + 5\sqrt{32} &= 3\sqrt{2} + 6\sqrt{2} + 5 \times 4\sqrt{2} \\ &= 3\sqrt{2} + 6\sqrt{2} + 20\sqrt{2} \\ &= 29\sqrt{2} \end{aligned}$$

So $k = 29$.

Exam­ple 2: divid­ing#

Sim­plify $\displaystyle \frac{\sqrt{108}}{\sqrt{45}}$.

$$\displaystyle \frac{\sqrt{108}}{\sqrt{45}} = \frac{6\sqrt{3}}{3\sqrt{5}} = \frac{2\sqrt{3}}{\sqrt{5}}$$

There is still a surd in the denom­i­na­tor. Mul­ti­ply top and bot­tom by $\sqrt{5}$, its ratio­nal­is­ing fac­tor from the last les­son:

$$\displaystyle \frac{2\sqrt{3}}{\sqrt{5}} = \frac{2\sqrt{3} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{2\sqrt{15}}{5}$$

Pow­ers of surds#

To raise a surd to a power, use $(\sqrt{a})^{2} = a$ as many times as you can. Split the power into pairs; each pair gives $a$, and any one left over stays as $\sqrt{a}$.

$$\begin{aligned} (\sqrt{2})^{3} &= (\sqrt{2})^{2} \times \sqrt{2} = 2\sqrt{2} \\ (\sqrt{2})^{4} &= (\sqrt{2})^{2} \times (\sqrt{2})^{2} = 4 \\ (\sqrt{2})^{5} &= \left[(\sqrt{2})^{2}\right]^{2} \cdot \sqrt{2} = 4\sqrt{2} \\ (\sqrt{2})^{11} &= \left[(\sqrt{2})^{2}\right]^{5} \cdot \sqrt{2} = 2^{5}\sqrt{2} = 32\sqrt{2} \\ (\sqrt{3})^{3} &= (\sqrt{3})^{2} \times \sqrt{3} = 3\sqrt{3} \\ (\sqrt{3})^{13} &= \left[(\sqrt{3})^{2}\right]^{6} \cdot \sqrt{3} = 3^{6}\sqrt{3} = 729\sqrt{3} \end{aligned}$$

An even power gives a whole num­ber; an odd power leaves one $\sqrt{a}$ behind.

For a mixed surd, raise each part to the power sep­a­rately:

$$\begin{aligned} (2\sqrt{2})^{2} &= 2^{2} \times (\sqrt{2})^{2} = 4 \times 2 = 8 \\ (2\sqrt{2})^{3} &= 2^{3} \times (\sqrt{2})^{3} = 8 \times 2\sqrt{2} = 16\sqrt{2} \end{aligned}$$

Exam­ple 3#

Find $(2\sqrt{2})^{5}$.

$$\begin{aligned} (2\sqrt{2})^{5} &= 2^{5} \cdot (\sqrt{2})^{5} \\ &= 32 \times 4\sqrt{2} \\ &= 128\sqrt{2} \end{aligned}$$

A com­mon wrong answer is $512\sqrt{2}$. Check it by writ­ing $2\sqrt{2} = \sqrt{8}$: then $(\sqrt{8})^{5} = 8^{2}\sqrt{8} = 64 \times 2\sqrt{2} = 128\sqrt{2}$.

Exam­ple 4#

Find $(3\sqrt{3})^{4}$.

$$\begin{aligned} (3\sqrt{3})^{4} &= 3^{4} \times (\sqrt{3})^{4} \\ &= 81 \times 9 \\ &= 729 \end{aligned}$$

Prac­tice#

1. Sim­plify $\sqrt{45}$.
2. Sim­plify $\sqrt{200}$.
3. Sim­plify $\sqrt{147}$.
4. Write $4\sqrt{3}$ as a pure surd.
5. Sim­plify $\sqrt{12} + \sqrt{27} + \sqrt{75}$.
6. Sim­plify $3\sqrt{50} - 2\sqrt{8}$.
7. If $\sqrt{98} + \sqrt{50} - \sqrt{32} = k\sqrt{2}$, find $k$.
8. Sim­plify $\displaystyle \frac{\sqrt{72}}{\sqrt{50}}$.
9. Find $(\sqrt{5})^{5}$.
10. Find $(3\sqrt{2})^{3}$.
11. Find $(\sqrt{2})^{8}$.

Answers#