Question

simplify the following radical expression by using the conjugate.

\\\frac{1}{7 - \sqrt{3}}\\

\\\frac{? + \sqrt{\quad}}{\quad}\\

Explanation:

Identify the conjugate of the denominator

Using the Rationalizing Denominators knowledge point
The denominator is \(7 - \sqrt{3}\). Its conjugate is \(7 + \sqrt{3}\).

Multiply the numerator and denominator by the conjugate

Using the Rationalizing Denominators knowledge point

$$ \frac{1}{7 - \sqrt{3}} \cdot \frac{7 + \sqrt{3}}{7 + \sqrt{3}} = \frac{7 + \sqrt{3}}{(7 - \sqrt{3})(7 + \sqrt{3})} $$

Simplify the denominator

Using the Rationalizing Denominators knowledge point

$$ (7 - \sqrt{3})(7 + \sqrt{3}) = 7^2 - (\sqrt{3})^2 = 49 - 3 = 46 $$

Write the simplified expression in the required format

Using the Rationalizing Denominators knowledge point

$$ \frac{7 + \sqrt{3}}{46} = \frac{7}{46} + \frac{\sqrt{3}}{46} $$

Comparing this to the target format:

$$ \frac{[?] + \sqrt{\quad}}{\quad} $$

We have:

$$ \frac{7 + \sqrt{3}}{46} $$

Thus, the value in the green box with the question mark \([?]\) is \(7\).

Answer:

Simplify the following radical expression by using the conjugate.

$$\frac{1}{7-\sqrt{3}}$$

The simplified expression is <blank>\(\frac{7 + \sqrt{3}}{46}\)</blank>, where the value in the green box \([?]\) is <blank>\(7\)</blank>.