Question

simplify the following radical expression by using the conjugate.

\\\frac{1}{\sqrt{26} + \sqrt{11}}\\

\\\frac{\sqrt{26} - \sqrt{?}}{\quad}\\

Explanation:

Identify the conjugate of the denominator

The denominator of the given expression is \(\sqrt{26} + \sqrt{11}\). Its conjugate is \(\sqrt{26} - \sqrt{11}\).

Multiply numerator and denominator by the conjugate

$$ \frac{1}{\sqrt{26} + \sqrt{11}} \cdot \frac{\sqrt{26} - \sqrt{11}}{\sqrt{26} - \sqrt{11}} = \frac{\sqrt{26} - \sqrt{11}}{(\sqrt{26})^2 - (\sqrt{11})^2} $$

Simplify the denominator

$$ \frac{\sqrt{26} - \sqrt{11}}{26 - 11} = \frac{\sqrt{26} - \sqrt{11}}{15} $$

Comparing this to the template \(\frac{\sqrt{26} - \sqrt{[?]}}{\text{blank}}\), the value inside the square root is \(11\).

Answer:

Simplify the following radical expression by using the conjugate.

$$\frac{1}{\sqrt{26} + \sqrt{11}}$$

$$\frac{\sqrt{26} - \sqrt{\text{11}}}{15}$$