Divide numeric surds

To simplify the expression $\frac{\sqrt{20}}{\sqrt{2}}$√20√2​ 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

 $\frac{a^{\frac{1}{2}}}{b^{\frac{1}{2}}}=\left(\frac{a}{b}\right)^{\frac{1}{2}}$a12​b12​​=(ab​)12​

Or written in surd form,

$\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$√a√b​=√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}}$4√182√3​.

Think: For convenience we divide the surd parts and the non-surd parts separately. If we use the rule $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$√a√b​=√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.

$\frac{\sqrt[k]{a}}{\sqrt[k]{b}}=\sqrt[k]{\frac{a}{b}}$^k√a^k√b​=^k√ab​

Example 2

Simplify the expression $\frac{\sqrt[3]{32}}{\sqrt[3]{4}}$³√32³√4​.

Think: Using the rule that $\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}$³√a³√b​=³√ab​ 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

Question 2

Question 3