Simplify. Rationalize the denominator.

Question

7
8+
3

Answer 1

To simplify the expression \( \frac{7}{8 + \sqrt{3}} \) and rationalize the denominator, follow these steps:

1. Multiply both the numerator and the denominator by the conjugate of the denominator \( 8 - \sqrt{3} \):

\[ \frac{7}{8 + \sqrt{3}} \cdot \frac{8 - \sqrt{3}}{8 - \sqrt{3}} = \frac{7(8 - \sqrt{3})}{(8 + \sqrt{3})(8 - \sqrt{3})} \]

2. Calculate the denominator using the difference of squares formula:

\[ (8 + \sqrt{3})(8 - \sqrt{3}) = 8^2 - (\sqrt{3})^2 = 64 - 3 = 61 \]

3. Calculate the numerator:

\[ 7(8 - \sqrt{3}) = 56 - 7\sqrt{3} \]

4. Combine everything:

\[ \frac{56 - 7\sqrt{3}}{61} \]

So, the simplified and rationalized form of the expression is:

\[ \frac{56 - 7\sqrt{3}}{61} \]