<s>[INST] How to convert $\sqrt{\frac{5}{3}}$ to $\frac{\sqrt{15}}{3}$? Disclosure: This is homework, but not part of the homework. This is just something that I do not understand.

$$ x = \sqrt{\frac{5}{3}} $$
$$ x = \frac{\sqrt{15}}{3} $$

Could anyone please explain this to me?
Thanks in advance.
 [/INST] $$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$$
Since $\frac{\sqrt{b}}{\sqrt{b}}=1$, then we can multiply the right hand side of the above expression by $\frac{\sqrt{b}}{\sqrt{b}}$, we get:
$$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}$$
Note that $\sqrt{a} \times \sqrt{b}=\sqrt{ab}$ , and note that $\sqrt{b} \times \sqrt{b}=\sqrt{b}^2=b$.
Therefore,
$$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}=\frac{\sqrt{ab}}{b}$$
In this case,
$$\sqrt{\frac{5}{3}}=\frac{\sqrt{5}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{5 \cdot 3}}{3}=\frac{\sqrt{15}}{3}$$
This question was asked long time before, but I answer it in his way because I think someone will see his answer and will get use of it.
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