# What is x if x^(-1/2)=5+sqrt(1/12)?

Oct 22, 2015

Calculated for every step so that you can see where everything comes from (long answer!)
$x = \frac{12}{301 + 20 \sqrt{3}}$

#### Explanation:

It is all about understanding manipulation and what things mean:

Given that: ${x}^{- \frac{1}{2}} = 5 + \sqrt{\frac{1}{12}}$............. ( 1 )

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First you need to understand that x^(-1/2) = 1/(sqrt(x)
You also need to know that $\sqrt{\frac{1}{12}} = \frac{\sqrt{1}}{\sqrt{12}} = \frac{1}{\sqrt{12}}$

So write ( 1 ) as:

$\frac{1}{\sqrt{x}} = 5 + \frac{1}{\sqrt{12}}$ ....... (2)

The thing is, we need to gat $x$ on its own. So we do everything we can to change $\frac{1}{\sqrt{x}}$ to just $x$.

First we need to get rid of the root. This can be done by squaring everything in (2) giving:

${\left(\frac{1}{\sqrt{x}}\right)}^{2} = {\left(5 + \frac{1}{\sqrt{12}}\right)}^{2}$

$\frac{1}{x} = {5}^{2} + \frac{10}{\sqrt{12}} + \frac{1}{12}$

Now we put all the right hand side over a common denominator

$\frac{1}{x} = \frac{\left(12 \times {5}^{2}\right) + \left(10 \times \sqrt{12}\right) + 1}{12}$

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But $12 \times {5}^{2} = 300$

$\sqrt{12} = \sqrt{3 \times 4} = 2 \sqrt{3}$
so $10 \sqrt{12} = 20 \sqrt{3}$
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Substitution gives:

$\frac{1}{x} = \frac{300 + 20 \sqrt{3} + 1}{12}$

We need $x$ on its own so we can simply turn everything upside down giving:

$x = \frac{12}{301 + 20 \sqrt{3}}$