Simplifying a fraction with radicals

Question

Simplify the following fraction as much as possible $$\def\A{\sqrt{15}}\def\B{\sqrt{35}} \frac{(8+2\A)^{3/2}+(8-2\A)^{3/2}}{(12+2\B)^{3/2}-(12-2\B)^{3/2}} $$

This problem is just driving me insane. I've tried so many different approaches but it keeps becoming a mess for me

I worked with conjugates but that won't help. The first step would be to get rid of the exponent.

Any help on this problem?

Answer 1

Hint:

• What is $\Big(\sqrt5 + \sqrt3\Big)^2$ and $\Big(\sqrt5 - \sqrt3\Big)^2$ ?

• What is $\Big(\sqrt7 + \sqrt5\Big)^2$ and $\Big(\sqrt7 - \sqrt5\Big)^2$ ?

Another hint:

Please find the following two formulas and use them.

1. $(a+b)^3 + (a-b)^3$

2. $(a+b)^3 - (a-b)^3$

Answer 2

For the nominator I get: $$\sqrt{8+2\sqrt{15}}^3+\sqrt{8-2\sqrt{15}}^3 = 28\sqrt{5}$$ For the denominator I get: $$\sqrt{12+2\sqrt{35}}^3-\sqrt{12-2\sqrt{35}}^3 = 52\sqrt{5}$$ So in the end I get: $$\frac{28\sqrt{5}}{52\sqrt{5}} = \frac{7}{13}$$