# DIVISION OF SURDS

As is with numbers, surds can also be divided.

The rule is this:
$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Example 1
Simplify $\frac{\sqrt{72}}{\sqrt{8}}$

Solution:
$\frac{\sqrt{72}}{\sqrt{8}}=\sqrt{\frac{72}{8}}$
$\sqrt{\frac{72}{8}}=\sqrt{9}$
$\sqrt{9}=3$

Example 2
Simplify $\frac{\sqrt{192}}{\sqrt{3}}$

Solution:
$\frac{\sqrt{192}}{\sqrt{3}}$ can be written as $\sqrt{\frac{192}{3}}$
$\sqrt{\frac{192}{3}}=\sqrt{64}$
$\sqrt{64}=8$

Example 3
Write $\frac{\sqrt{3}\times\sqrt{18}}{\sqrt{18}}$ in the simplest form possible.

Solution:
$\frac{\sqrt{3}\times\require{cancel}\bcancel{\sqrt{18}}}{\require{cancel}\bcancel{\sqrt{18}}}$
$\sqrt{3}$

In some cases, you may need to rationalize the surd. Rationalizing the surd is the processing of removing the irrational number from the denominator of the surd.

To rationalize a surd, you quickly multiply the denominator and numerator by the conjugates of the denominator so that the denominator becomes a rational number

Example 4
By rationalizing the denominator, simplify $\frac{12}{\sqrt{6}}$

Solution
Rationalizing the denominator
$\frac{12}{\sqrt{6}}\times\frac{\sqrt{6}}{\sqrt{6}}$
$\frac{12\sqrt{6}}{\sqrt{36}}$
$\frac{12\sqrt{6}}{6}$

If $\frac{12}{6}=2$, then
$\frac{12\sqrt{6}}{6}=2\sqrt{6}$

Note: the multiplication of $\frac{\sqrt{6}}{\sqrt{6}}$ is equivalent to multiplication by $1$. Hence the value of the given fraction($\frac{12}{\sqrt{6}})$ correspond with our derived answer ($2\sqrt{6}$).

Related post
Example 5
By Rationalizing the denominator, write $\frac{20\sqrt{2}}{\sqrt{12}}$ in the simplest form possible.

Solution
$\sqrt{12}$ can be simplified further, hence
$\frac{20\sqrt{2}}{\sqrt{4\times3}}$
$\frac{20\sqrt{2}}{2\sqrt{3}}$

Now, let's rationalize the denominator
$\frac{20\sqrt{2}}{2\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}$
$\frac{20\sqrt{6}}{2\sqrt{9}}$
$\frac{20\sqrt{6}}{2(3)}$
$\frac{20\sqrt{6}}{6}$
$\frac{10\sqrt{6}}{3}$

That is all for now. We would be concluding our series on surds in this next post. We would expect your question in our telegram community.
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