Question

How do I do this step by step?

Answer 1

`\large\boxed{(2\sqrt3)/(2+3\sqrt2)=(3\sqrt6)/(7)-(2\sqrt3)/(7)}`

Explanation:

`(2\sqrt3)/(2+3\sqrt2)\\\\\text{use}\ (a+b)(a-b)=a^2-b^2\\\\=(2\sqrt3)/(2+3\sqrt2)\cdot(2-3\sqrt2)/(2-3\sqrt2)=(2\sqrt3(2-3\sqrt2))/(2^2-(3\sqrt2)^2)\\\\\text{use the distributive property}\ a(b+c)=ab+ac\\\\=((2\sqrt3)(2)+(2\sqrt3)(-3\sqrt2))/(4-3^2(\sqrt2)^2)\\\\\text{use}\ (√(a))^2=a\ \text{and}\ √(a)\cdot√(b)=√(ab)\\\\=(4\sqrt3-6\sqrt6)/(4-(9)(2))=(4\sqrt3-6\sqrt6)/(4-18)=(4\sqrt3-6\sqrt6)/(-14)=-(4\sqrt3)/(14)+(6\sqrt6)/(14)\\\\=(3\sqrt6)/(7)-(2\sqrt3)/(7)`