To simplify the expression [tex]\(\frac{5}{4 - \sqrt{11}}\)[/tex] by rationalizing the denominator, we follow these steps:
1. Identify the conjugate of the denominator: The denominator is [tex]\(4 - \sqrt{11}\)[/tex]. Its conjugate is [tex]\(4 + \sqrt{11}\)[/tex].
2. Multiply the numerator and denominator by the conjugate: This will help eliminate the square root in the denominator.
[tex]\[
\frac{5}{4 - \sqrt{11}} \times \frac{4 + \sqrt{11}}{4 + \sqrt{11}}
\][/tex]
3. Distribute in the numerator: Multiply the numerator [tex]\(5\)[/tex] by the conjugate [tex]\(4 + \sqrt{11}\)[/tex]:
[tex]\[
5 \times (4 + \sqrt{11}) = 5 \times 4 + 5 \times \sqrt{11} = 20 + 5\sqrt{11}
\][/tex]
So, the new numerator is [tex]\(20 + 5\sqrt{11}\)[/tex].
4. Simplify the denominator using the difference of squares: The denominator, when multiplied by its conjugate, uses the difference of squares formula [tex]\(a^2 - b^2\)[/tex]. Here, [tex]\(a = 4\)[/tex] and [tex]\(b = \sqrt{11}\)[/tex]:
[tex]\[
(4 - \sqrt{11})(4 + \sqrt{11}) = 4^2 - (\sqrt{11})^2 = 16 - 11 = 5
\][/tex]
So, the simplified denominator is [tex]\(5\)[/tex].
5. Put the new numerator over the new denominator:
[tex]\[
\frac{20 + 5\sqrt{11}}{5}
\][/tex]
6. Simplify the fraction: We can distribute the denominator over each term in the numerator:
[tex]\[
\frac{20}{5} + \frac{5\sqrt{11}}{5} = 4 + \sqrt{11}
\][/tex]
So, the simplified expression for [tex]\(\frac{5}{4 - \sqrt{11}}\)[/tex] is [tex]\(4 + \sqrt{11}\)[/tex].
When calculating the numerical result, we find:
[tex]\[
\frac{5}{4 - \sqrt{11}} \approx 7.3166247903554
\][/tex]
Thus, the detailed step-by-step solution confirms that the result is approximately [tex]\(7.3166247903554\)[/tex].
