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

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?

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$$\frac{1}{7\sqrt{7}}\cdot \frac{\sqrt{7}}{\sqrt{7}}.$$ – Daniel Fischer Dec 8 '13 at 14:52
Multiply both numerator and denominator bt Sqrt[7] and simplify the denominator [7 Sqrt(7) Sqrt(7)] = 7 7 = 49 – Claude Leibovici Dec 8 '13 at 14:52
If you replace 7 by x, you have the same solution – Claude Leibovici Dec 8 '13 at 15:04

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

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

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

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It is just an arithmetic. Just multiply the denominator and numerator by x. Then you're done.

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This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. – Dennis Gulko Dec 8 '13 at 15:57
@DennisGulko: This seems to be about the same as another answer. – robjohn Dec 8 '13 at 23:19