Question

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

Answer 1

Calculated for every step so that you can see where everything comes from (long answer!)
`x= (12)/(301+20sqrt(3))`

Explanation:

It is all about understanding manipulation and what things mean:

Given that: `x^(-1/2)= 5 + sqrt(1/12)`............. ( 1 )

.¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
First you need to understand that `x^(-1/2) = 1/(sqrt(x)`
You also need to know that `sqrt(1/12) = (sqrt(1))/(sqrt(12)) = 1/(sqrt(12))`

So write ( 1 ) as:

`1/(sqrt(x)) = 5 + 1/(sqrt(12))` ....... (2)

The thing is, we need to gat `x` on its own. So we do everything we can to change `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:

`(1/(sqrt(x)))^2 = (5+ 1/(sqrt(12)))^2`

`1/x = 5^2 + (10)/(sqrt(12)) + 1/12`

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

`1/x =( (12 times 5^2) + (10 times sqrt(12)) + 1 )/12`

.¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
But `12 times 5^2 = 300`

`sqrt(12) = sqrt(3 times 4) = 2sqrt(3)`
so `10sqrt(12) = 20sqrt(3)`
.¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

Substitution gives:

`1/x = (300 +20sqrt(3) +1)/12`

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

`x= (12)/(301+20sqrt(3))`