Question

Rationalize the denominator and simplify.

$\frac{10}{\sqrt{2}-1}$

Short answer

Rationalized form of the given expression is$10\sqrt{2}+10$.

Step-by-step solution

Step 1. To determine the conjugate if present. Otherwise simply eliminate the root terms using multiplication on numerator and denominator.

$\frac{10}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}$

Step 2. Re write the expression after multiplying.

$\frac{10}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}=\frac{10\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}$

Step 3. Simplify the expression.

$\begin{array}{c}\frac{10\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}=\frac{10\left(\sqrt{2}+1\right)}{\left({\left(\sqrt{2}\right)}^2-1^2\right)}\\ =\frac{10\sqrt{2}+10}{2-1}\\ =10\sqrt{2}+10\end{array}$