Question

Simplify
${(\sqrt{x}+\frac{1}{\sqrt{x}})}^3$

Answer 1

${(\sqrt{x}+\frac{1}{\sqrt{x}})}^3$ is of the form

${(a+b)}^3=a^3+b^3+3ab(a+b)$

where $a=\sqrt{x},b=\frac{1}{\sqrt{x}}$

$={\left(\sqrt{x}\right)}^3+{\left(\frac{1}{\sqrt{x}}\right)}^3+3\sqrt{x}\times \frac{1}{\sqrt{x}}(\sqrt{x}+\frac{1}{\sqrt{x}})$

$=x\sqrt{x}+\frac{1}{x\sqrt{x}}+3\sqrt{x}+\frac{3}{\sqrt{x}}$

$=(x+3)\sqrt{x}+(\frac{1}{x}+3)\frac{1}{\sqrt{x}}$