To rationalize the denominator of [tex]\(\frac{26}{4 + \sqrt{3}}\)[/tex], we need to eliminate the square root from the denominator. To achieve this, we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(4 + \sqrt{3}\)[/tex] is [tex]\(4 - \sqrt{3}\)[/tex]. Here are the steps:
1. Multiply by the conjugate:
[tex]\[
\frac{26}{4 + \sqrt{3}} \times \frac{4 - \sqrt{3}}{4 - \sqrt{3}}
\][/tex]
2. Numerator multiplication:
- Expand the numerator:
[tex]\[
26 \times (4 - \sqrt{3}) = 26 \times 4 - 26 \times \sqrt{3} = 104 - 26\sqrt{3}
\][/tex]
3. Denominator multiplication:
- Use the difference of squares formula [tex]\( (a + b)(a - b) = a^2 - b^2\)[/tex]:
[tex]\[
(4 + \sqrt{3})(4 - \sqrt{3}) = 4^2 - (\sqrt{3})^2 = 16 - 3 = 13
\][/tex]
4. Rewrite the fraction:
[tex]\[
\frac{104 - 26\sqrt{3}}{13}
\][/tex]
5. Simplify the fraction:
- Divide each term in the numerator by the denominator:
[tex]\[
\frac{104}{13} - \frac{26\sqrt{3}}{13} = 8 - 2\sqrt{3}
\][/tex]
Thus, the rationalized form of [tex]\(\frac{26}{4 + \sqrt{3}}\)[/tex] is:
[tex]\[
8 - 2\sqrt{3}
\][/tex]
