Answer
$\frac{\sqrt{ab}}b$
Work Step by Step
We have to simplify the following radical expression:
$$\frac{a\sqrt b}{b\sqrt a}$$
To simplify a radical expression, all the following statements must be true:
1. The radical has no perfect square factors other than 1.
2. The radical contains no functions.
3. No radicals appear in the denominator of a fraction.
Since we do not know the values of $a$ and $b$, we cannot tell if the first and second requirements are met. However, we can see that the fraction has a radical in the denominator. To rationalize it, let's multiply the fraction by $\frac{\sqrt a}{\sqrt a}$.
$$\frac{a\sqrt b}{b\sqrt a} \times \frac{\sqrt a}{\sqrt a} = \frac{a\sqrt b \times \sqrt a}{b\sqrt a \times \sqrt a}$$
$\frac{\sqrt a}{\sqrt a} = 1$, so the value of our expression remains unchanged. Next, we can apply the multiplication property of square roots, which states:
If $a\geq0$ and $b\geq0$, then $\sqrt{ab} = \sqrt a \times \sqrt b$
Let's use this property and simplify the expression as much as we can.
$\frac{a\sqrt b\times \sqrt a}{b\sqrt a \times\sqrt a}$
$\frac{a\sqrt{ab}}{b\sqrt{a^2}}$
$\frac{a\times\sqrt{ab}}{a\times b}$ | cancel out the $a$
$\frac{\sqrt{ab}}{b}$
Now the fraction does not have a radical in the denominator, so it is as simplified as possible.
