Write $\sqrt{\dfrac{a}{5b}}$ without denominator under the radical sign

Question

I am supposed to write the irrational algebraic expression without denominator under the radical sign. $$\sqrt{\dfrac{a}{5b}};ab>0$$

The given answer in my book is $\dfrac{1}{5|b|}\sqrt{5ab}$

My first try was to write the given expression as $$\sqrt{\dfrac{a}{5b}}=\dfrac{\sqrt{a}}{\sqrt{5b}}.\dfrac{\sqrt{5b}}{\sqrt{5b}}=\dfrac{\sqrt{5ab}}{5b}$$ I realised I am wrong, though, because we aren't sure that $\sqrt{a}$ and $\sqrt{5b}$ exist in $\mathbb{R}$ as we only have $ab>0$ which means that $a$ and $b$ can both be negative. How should we solve the problem then?

Answer 1

You have$$\sqrt\frac{a}{5b}=\sqrt\frac{5ab}{5^2b^2}=\frac{\sqrt{5ab}}{5|b|}.$$

Answer 2

$$\sqrt{\frac{a}{5b}}=\sqrt{\frac{a}{5b}\frac{5b}{5b}}=\frac{\sqrt{5ab}}{\sqrt{5^2b^2}}=\frac{1}{5|b|}\sqrt{5ab}$$

Answer 3

the problem is when $a$ and $b$ are both negative, in which case we replace them both by $\mid a \mid$ and $\mid b \mid$, without changing the value of the square root, since we multiplied what is inside it by $\dfrac{-1}{-1}$, and then you can use your first idea. This reaonning hold when $a$ and $b$ are poth positive.