Powers and Roots – Simplify an expression with roots – Exercise 5679

Exercise

Simplify the expression:

$\frac{1}{\sqrt[4]{2}+1}$

Final Answer

Show final answer

Solution

$\frac{1}{\sqrt[4]{2}+1}=$

$=\frac{1}{2^{\frac{1}{4}}+1}=$

$=\frac{1}{{(\sqrt{2})}^2+1}=$

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

$=\frac{{(\sqrt{2})}^2-1}{\sqrt{2}-1}=$

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

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

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

$=(\sqrt[4]{2}-1)(\sqrt{2}+1)$