Question

Me ajuda glr! ahh, explica como faz também! ;)

Answer 1

O m.m.c entre os denominadores é o produto desses: [(√n) + 1][(√n) - 1]

$\dfrac{\sqrt{n}}{\sqrt{n}+1}+\dfrac{1}{\sqrt{n}-1}=\dfrac{\sqrt{n}(\sqrt{n}-1)}{(\sqrt{n}+1)(\sqrt{n}-1)}+\dfrac{1(\sqrt{n}+1)}{(\sqrt{n}+1)(\sqrt{n}-1)}\\\\\\\dfrac{\sqrt{n}}{\sqrt{n}+1}+\dfrac{1}{\sqrt{n}-1}=\dfrac{n-\sqrt{n}}{(\sqrt{n})^{2}-1^{2}}+\dfrac{\sqrt{n}+1}{(\sqrt{n})^{2}-1^{2}}\\\\\\\dfrac{\sqrt{n}}{\sqrt{n}+1}+\dfrac{1}{\sqrt{n}-1}=\dfrac{n-\sqrt{n}}{n-1}+\dfrac{\sqrt{n}+1}{n-1}$

Somando as frações:

$\dfrac{\sqrt{n}}{\sqrt{n}+1}+\dfrac{1}{\sqrt{n}-1}=\dfrac{n-\sqrt{n}+\sqrt{n}+1}{n-1}\\\\\\\boxed{\boxed{\dfrac{\sqrt{n}}{\sqrt{n}+1}+\dfrac{1}{\sqrt{n}-1}=\dfrac{n+1}{n-1}}}$