Question

(√a+x)-(√a-x)÷ (√a+x)+(√a-x)​

Answer 1

Explanation:

$\mathsf{Let\:a+x\:be\:p\:and\:a-x\:be\:q.}$

$\mathsf{\frac{\sqrt{p}-\sqrt{q}}{{\sqrt{p}+\sqrt{q}}}=\frac{(\sqrt{p}-\sqrt{q})(\sqrt{p}-\sqrt{q})}{(\sqrt{p}+\sqrt{q})(\sqrt{p}-\sqrt{q})}}$

$\mathsf{=\frac{(\sqrt{p}-\sqrt{q})^2}{{(\sqrt{p)^2-(\sqrt{q})^2}}}}$

$\mathsf{=\frac{\sqrt{p}^2-2\sqrt{pq}+\sqrt{q}^2}{p-q}}$

$\mathsf{=\frac{p-2\sqrt{pq}+q}{p-q}}$

$\mathsf{=\frac{a+x-2\sqrt{(a+x)(a-x)}+a-x}{a+x-a+x}}$

$\mathsf{=\frac{2a-2\sqrt{a^2-x^2}}{2x}}$

$\mathsf{=\frac{a-\sqrt{a^2-x^2}}{x}}$

This is the answer to the rationalization question.

Answer: a-√(a^2 - x^2)/x

I hope this is helpful and is the correct answer. :D