# How do you divide (2sqrt(3))/(2sqrt(5)) and sqrt(5)/sqrt(10)?

Mar 22, 2015

Ok, by divide, I presume you are talking about rationalising the denominator.

For info on how to do this, check this site out: http://www.purplemath.com/modules/radicals5.htm

Ok, on to solving:

In the first one, notice how the problem can be rewritten as: $\frac{2 \cdot \sqrt{3}}{2 \cdot \sqrt{5}}$. This can be rewritten as $\left(\frac{2}{2}\right) \cdot \left(\frac{\sqrt{3}}{\sqrt{5}}\right)$. Since $\frac{2}{2}$ is just 1, we can rewrite the problem as

$\left(1\right) \cdot \left(\frac{\sqrt{3}}{\sqrt{5}}\right)$, which is just $\left(\frac{\sqrt{3}}{\sqrt{5}}\right)$.

From here, we rationalise the denominator by multiplying the expression by $\frac{\sqrt{5}}{\sqrt{5}}$. We can do this because $\frac{\sqrt{5}}{\sqrt{5}}$ is simply 1, and multiplying something by 1 doesn't change the nature of the expression.

So our expression becomes: $\frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$, which simplifies to become $\frac{\sqrt{15}}{5}$. Since $\sqrt{15}$ is not something we can simplify, our final answer remains $\frac{\sqrt{15}}{5}$.

Now for the second problem, the procedure is basically the same.

We multiply the expression by $\frac{\sqrt{10}}{\sqrt{10}}$, so we get: $\frac{\sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$, which simplifies to become $\frac{\sqrt{50}}{\sqrt{100}}$.

Since $\sqrt{100}$ simplifies to 10, the expression can be simplified to read: $\frac{\sqrt{50}}{10}$.

Now unlike the last problem, this numerator can be simplified, as it is a multiple of 25. $\sqrt{50} = 5 \sqrt{2}$

So our expression reads $\frac{5 \sqrt{2}}{10}$, which can be simplified to $\frac{\sqrt{2}}{2}$ for our final answer.