Question

rationalize the denominator.
\\(\frac{7}{1 + \sqrt{5}}\\)

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of \(1 + \sqrt{5}\) is \(1 - \sqrt{5}\). So we multiply the fraction \(\frac{7}{1 + \sqrt{5}}\) by \(\frac{1 - \sqrt{5}}{1 - \sqrt{5}}\):

$$ \frac{7(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})} $$

Step2: Simplify the denominator using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\).

Here, \(a = 1\) and \(b=\sqrt{5}\), so the denominator becomes \(1^2 - (\sqrt{5})^2 = 1 - 5=-4\). The numerator is \(7(1 - \sqrt{5}) = 7 - 7\sqrt{5}\). So the fraction is now:

$$ \frac{7 - 7\sqrt{5}}{-4} $$

Step3: Simplify the fraction by factoring out the negative sign from the denominator (or multiplying numerator and denominator by - 1).

$$ \frac{-(7\sqrt{5}-7)}{-4}=\frac{7\sqrt{5}-7}{4} $$

Answer:

\(\frac{7\sqrt{5}-7}{4}\) (or equivalently \(\frac{7( \sqrt{5}-1)}{4}\))