Rationalize the denominator and simplify. $$ \frac{\sqrt{7}}{14-\sqrt{7}} $$

Question

UpStudy AI · Accepted Answer

To rationalize the denominator and simplify the expression:

$$
\frac{\sqrt{7}}{14 - \sqrt{7}}
$$

**Step 1: Multiply by the Conjugate**

Multiply the numerator and the denominator by the conjugate of the denominator, which is $$14 + \sqrt{7}$$:

$$
\frac{\sqrt{7}}{14 - \sqrt{7}} \times \frac{14 + \sqrt{7}}{14 + \sqrt{7}} = \frac{\sqrt{7}(14 + \sqrt{7})}{(14 - \sqrt{7})(14 + \sqrt{7})}
$$

**Step 2: Simplify the Denominator**

Use the difference of squares formula $$ (a - b)(a + b) = a^2 - b^2 $$:

$$
(14 - \sqrt{7})(14 + \sqrt{7}) = 14^2 - (\sqrt{7})^2 = 196 - 7 = 189
$$

**Step 3: Expand the Numerator**

$$
\sqrt{7}(14 + \sqrt{7}) = 14\sqrt{7} + 7
$$

**Step 4: Combine and Simplify**

$$
\frac{14\sqrt{7} + 7}{189} = \frac{7(2\sqrt{7} + 1)}{7 \times 27} = \frac{2\sqrt{7} + 1}{27}
$$

**Final Answer:**

$$
\frac{2\sqrt{7} + 1}{27}
$$
$$ \frac{2\sqrt{7} + 1}{27} $$

Rationalize the denominator and simplify.