# Rationalize the denominator and simplify

So the problem I have says rationalize the denominator and simplify. $$\frac{ \sqrt{15}}{\sqrt{10}-3}$$

My answer I got was $\frac{5 \sqrt6}{7}$.

Am I doing this wrong or is this the wrong answer I was told it was incorrect?

• So it is pretty easy to check that your answer is wrong, just use a calculator. You will get $23.87$ for the question and $1.75$ for your answer. – David Aug 27 '14 at 0:13

It seems that you tried multiplying by $\frac{\sqrt{10}}{\sqrt{10}}$. Instead, you should try multiplying by the conjugate and take advantage of difference of squares: $$\frac{\sqrt{15}}{\sqrt{10} - 3} = \frac{\sqrt{15}}{\sqrt{10} - 3} \cdot \frac{\sqrt{10} + 3}{\sqrt{10} + 3} = \frac{\sqrt{150} + 3\sqrt{15}}{(\sqrt{10})^2 - 3^2} = 5\sqrt{6} + 3\sqrt{15}$$
I think wht happened was that you correctly multiplied the denominator by $\sqrt{10}+3$, but incorrectly multiplied the numerator by $\sqrt{10}$. The numerator should also have been multiplied by $\sqrt{10}+3$
$$\frac{\sqrt{15}}{\sqrt{10}-3}=\frac{\sqrt{15} \cdot (\sqrt{10}+3)}{(\sqrt{10}-3) \cdot (\sqrt{10}+3)}=\frac{\sqrt{15} \cdot (\sqrt{10}+3)}{10-9}=\sqrt{15} \cdot (\sqrt{10}+3)$$