Question

$\frac{1}{\sqrt{9}-\sqrt{8}}$ is equal to:

• A. $\frac{1}{2}(3-2\sqrt{2})$
• B. $\frac{1}{3+2\sqrt{2}}$
• C. $3-2\sqrt{2}$
• D. $3+2\sqrt{2}$

Answer 1

The correct option is D

$3+2\sqrt{2}$

Explanation for the correct option:

The given term is$\frac{1}{\sqrt{9}-\sqrt{8}}$.

We need to find the value of it.

Simplify it by multiplying the numerator and the denominator by the conjugate $\left(\sqrt{9}+\sqrt{8}\right)$ of this term.

So, $\frac{1}{\sqrt{9}-\sqrt{8}}\times \frac{\sqrt{9}+\sqrt{8}}{\sqrt{9}+\sqrt{8}}$

Simplifying further,

$\frac{\sqrt{9}+\sqrt{8}}{{\left(\sqrt{9}\right)}^2-{\left(\sqrt{8}\right)}^2}$ Using this identity $\left(a^2-b^2\right)=\left(a+b\right)\left(a-b\right)$ in the denominator

$=\frac{\sqrt{9}+\sqrt{8}}{9-8}$ As we know that ${\left(\sqrt{9}\right)}^2=9\mathrm{and}{\left(\sqrt{8}\right)}^2=8$

$=\frac{\sqrt{9}+\sqrt{8}}{1}$

$=3+2\sqrt{2}$

So, the value of the given term $\frac{1}{\sqrt{9}-\sqrt{8}}$ is $3+2\sqrt{2}$.

Hence option (d) is correct.