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

1 Answer
Oct 22, 2015

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 )

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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#

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But #12 times 5^2 = 300#

#sqrt(12) = sqrt(3 times 4) = 2sqrt(3) #
so #10sqrt(12) = 20sqrt(3)#
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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))#