# Divide numeric surds

Lesson

To simplify the expression $\frac{\sqrt{20}}{\sqrt{2}}$202 we can find a useful result from the index laws.

We know from the index laws that $\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n$anbn=(ab)n. And we know that the index for a square root is $\frac{1}{2}$12. So if we substitute in $n=\frac{1}{2}$n=12 we get

$\frac{a^{\frac{1}{2}}}{b^{\frac{1}{2}}}=\left(\frac{a}{b}\right)^{\frac{1}{2}}$a12b12=(ab)12

Or written in surd form,

$\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$ab=ab

We can use this result to simplify our expression

 $\frac{\sqrt{20}}{\sqrt{2}}$√20√2​ $=$= $\sqrt{\frac{20}{2}}$√202​ (Simplifying into one surd) $=$= $\sqrt{10}$√10 (Simplifying the division under the surd)

This can also be extended if we used division as notation in the form

 $\sqrt{20}\div\sqrt{2}$√20÷​√2 $=$= $\sqrt{20\div2}$√20÷2 (Simplifying into one surd) $=$= $\sqrt{10}$√10 (Simplifying the division under the surd)

#### Worked example

##### Example 1

Simplify the expression $\frac{4\sqrt{18}}{2\sqrt{3}}$41823.

Think: For convenience we divide the surd parts and the non-surd parts separately. If we use the rule $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$ab=ab, we can then simplify the fraction under the square root symbol.

Do:

 $\frac{4\sqrt{18}}{2\sqrt{3}}$4√182√3​ $=$= $\frac{4}{2}\frac{\sqrt{18}}{\sqrt{3}}$42​√18√3​ (Separating division into surd and non-surd parts) $=$= $\frac{4}{2}\sqrt{\frac{18}{3}}$42​√183​ (Simplifying surds into one surd) $=$= $2\sqrt{6}$2√6 (Simplifying the two divisions)

### Roots of different degrees

Going back to the index law $\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n$anbn=(ab)n, we can see that this will work not just for $n=\frac{1}{2}$n=12 but also $n=\frac{1}{3},\frac{1}{4},\frac{1}{5}...$n=13,14,15... and so on. That is, this will also work with cube roots, fourth roots, etc, as long as both roots have the same value of $n$n - that is, the same degree.

$\frac{\sqrt[k]{a}}{\sqrt[k]{b}}=\sqrt[k]{\frac{a}{b}}$kakb=kab

##### Example 2

Simplify the expression $\frac{\sqrt[3]{32}}{\sqrt[3]{4}}$33234.

Think: Using the rule that $\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}$3a3b=3ab we can simplify the expression.

Do:

 $\frac{\sqrt[3]{32}}{\sqrt[3]{4}}$3√323√4​ $=$= $\sqrt[3]{\frac{32}{4}}$3√324​ (Simplifying into one surd) $=$= $\sqrt[3]{8}$3√8 (Simplifying division under the surd) $=$= $2$2 (Simplifying the cube root expression)

#### Practice questions

##### Question 1

Simplify the expression $\frac{\sqrt{143}}{\sqrt{11}}$14311.

##### Question 2

Simplify the expression $\frac{\sqrt[3]{40}}{\sqrt[3]{5}}$34035.

##### Question 3

Simplify the expression $\frac{21\sqrt{80}}{3\sqrt{5}}$218035.