To rationalize the denominator of the fraction [tex]\(\frac{7}{1 - \sqrt{5}}\)[/tex], we can use the method of multiplying by the conjugate of the denominator. Here’s the step-by-step process:
1. Identify the conjugate:
The conjugate of [tex]\(1 - \sqrt{5}\)[/tex] is [tex]\(1 + \sqrt{5}\)[/tex].
2. Multiply the numerator and the denominator by the conjugate:
[tex]\[
\frac{7}{1 - \sqrt{5}} \times \frac{1 + \sqrt{5}}{1 + \sqrt{5}} = \frac{7(1 + \sqrt{5})}{(1 - \sqrt{5})(1 + \sqrt{5})}
\][/tex]
3. Simplify the denominator using the difference of squares formula:
The difference of squares formula states that [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex].
Here, [tex]\(a = 1\)[/tex] and [tex]\(b = \sqrt{5}\)[/tex].
So,
[tex]\[
(1 - \sqrt{5})(1 + \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4
\][/tex]
4. Simplify the numerator:
Multiply the numerator by the conjugate:
[tex]\[
7(1 + \sqrt{5}) = 7 + 7\sqrt{5}
\][/tex]
5. Combine the results:
We now have:
[tex]\[
\frac{7(1 + \sqrt{5})}{-4} = \frac{7 + 7\sqrt{5}}{-4} = -\frac{7 + 7\sqrt{5}}{4}
\][/tex]
Therefore, the fraction [tex]\(\frac{7}{1 - \sqrt{5}}\)[/tex] can be rewritten as [tex]\( -\frac{7 + 7\sqrt{5}}{4} \)[/tex] when rationalized.
Hence, the correct answer is:
C. [tex]\( -\frac{7 + 7\sqrt{5}}{4} \)[/tex]
