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New Zealand
Level 8 - NCEA Level 3

Divide numeric surds


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


Or written in surd form,


We can use this result to simplify our expression

$\frac{\sqrt{20}}{\sqrt{2}}$202 $=$= $\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.


$\frac{4\sqrt{18}}{2\sqrt{3}}$41823 $=$= $\frac{4}{2}\frac{\sqrt{18}}{\sqrt{3}}$42183 (Separating division into surd and non-surd parts)
  $=$= $\frac{4}{2}\sqrt{\frac{18}{3}}$42183 (Simplifying surds into one surd)
  $=$= $2\sqrt{6}$26 (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.


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.


$\frac{\sqrt[3]{32}}{\sqrt[3]{4}}$33234 $=$= $\sqrt[3]{\frac{32}{4}}$3324 (Simplifying into one surd)
  $=$= $\sqrt[3]{8}$38 (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.




Manipulate complex numbers and present them graphically


Apply the algebra of complex numbers in solving problems

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