(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

sqrt(7 + 2sqrt(6)) - sqrt(7 - 2sqrt(6))

2. Relevant equations

3. The attempt at a solution

In order to work out the radicals, I define the answer as "x". Now I can square both sides to get rid of some of the first radicals:

sqrt(7 + 2sqrt(6)) - sqrt(7 - 2sqrt(6)) = x

(sqrt(7 + 2sqrt(6)) - sqrt(7 - 2sqrt(6)))^{2}= (x)^{2}

Now, I hope this is the right way of starting on this problem, continuing we get:

(7 + 2sqrt(6)) - 2 * sqrt(7+ 2 sqrt(6)) * sqrt(7 - 2 sqrt(6)) + (7 - 2 sqrt(6))

this implies:

14 - 2 * sqrt(7+ 2 sqrt(6)) * sqrt(7 - 2 sqrt(6)) = x^{2}

on moving the 14 to the RHS and squaring once more to get rid of more radicals:

(-2 * sqrt(7+ 2 sqrt(6)) * sqrt(7 - 2 sqrt(6)) )^{2}= (x^{2}-14)^{2}

this implies

4 * (7 + 2 sqrt(6)) * (7 - 2 sqrt(6)) = (x^{2}-14)^{2}

because the two factors with sqrts are conjugate pairs, the LHS will look like this:

4 * (7^{2}- (4*6) = 4 * 25 = 100

thus

100 = (x^{2}-14)^{2}

by taking the square root of both sides:

±10 = ± (x^{2}-14)

±10±14 = ± x^{2}

Now, this does not look like a proper solution. Working further I can get a couple of answers for x, but no way to check them or anything:

sqrt(24) = 2 sqrt(6) = ± x

sqrt(4) = 2 = ± x

Where do I go wrong?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Q3 - Radicals within radicals

**Physics Forums | Science Articles, Homework Help, Discussion**