Rationalizing the denominator and simplifying $\frac{ \sqrt{15}}{\sqrt{10}-3}$

Question

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

I got $\dfrac{5 \sqrt6}{7}$ for my answer.

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

Answer 1

$$\frac{\sqrt{15}}{\sqrt{10}-3}=\frac{\sqrt{15} \cdot (\sqrt{10}+3)}{(\sqrt{10}-3) \cdot (\sqrt{10}+3)}=\frac{\sqrt{15} \cdot (\sqrt{10}+3)}{10-9}=\sqrt{15} \cdot (\sqrt{10}+3)$$

Answer 2

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: $$ \frac{\sqrt{15}}{\sqrt{10} - 3} = \frac{\sqrt{15}}{\sqrt{10} - 3} \cdot \frac{\sqrt{10} + 3}{\sqrt{10} + 3} = \frac{\sqrt{150} + 3\sqrt{15}}{(\sqrt{10})^2 - 3^2} = 5\sqrt{6} + 3\sqrt{15} $$

Answer 3

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 $\sqrt{10}+3$