# Rationalizing complex root

1. Nov 9, 2015

### terryds

1. The problem statement, all variables and given/known data

If $\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=a+b\sqrt{6}$, then a+b is ?

A) 8
B) 7
C) 6
D) 5
E) 4

3. The attempt at a solution

This is my attempt:
$\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=\sqrt{\frac{10+4\sqrt{6}*(10+4\sqrt{6})}{10-4\sqrt{6}*(10+4\sqrt{6})}}=\sqrt{\frac{196+80\sqrt{6}}{4}}=\sqrt{49+20\sqrt{6}}$

Then, I got stuck.. I have no idea how to convert to form a+b√6

2. Nov 9, 2015

### Staff: Mentor

Square both sides of your original equation and then rationalize the denominator on the left side.
You should get something like m + n√(6) on one side and r + s√(6) on the other side. You can equate m with r and n with s to get two equations involving a and b.

3. Nov 9, 2015

### ehild

You have to use parentheses. The correct form is $\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=\sqrt{\frac{(10+4\sqrt{6})*(10+4\sqrt{6})}{(10-4\sqrt{6})*(10+4\sqrt{6})}}$
Is not the numerator the square of something? What is its square root?

4. Nov 9, 2015

### terryds

Thanks.. a is 5 and b is 2 , so a+b is 7..