Why does $ \frac{1}{x\sqrt{x}}= \frac {\sqrt{x}}{x^2} $?

Question

A homework question recently asked for me to simplify:

$\frac{1}{\sqrt{7}} \div {7}$

It's easy to see that this becomes

$\frac{1}{7\sqrt{7}}$

But according to wolfram alpha this is also equal to $\frac{\sqrt{7}}{49}$.

What sequence of steps can I use to get the second representation of this quantity from the first?

Answer 1

$\dfrac{1}{\sqrt{7}}\div7=\dfrac{1}{7\sqrt{7}}=\dfrac{1}{7\sqrt{7}}\cdot\dfrac{\sqrt{7}}{\sqrt{7}}=\dots$

Answer 2

$$\frac{1}{7\sqrt{7}} =\frac{\sqrt{7}}{7\sqrt{7}\cdot\sqrt{7}}=\frac{\sqrt{7}}{7\cdot7}=\frac{\sqrt{7}}{49}$$

Answer 3

This is called denominator rationalization. You multiply numerator and denominator with the same root, so you effectively move from denominator into the numerator.