Question

3 find the reduced radical form of \\(\dfrac{7}{1 - \sqrt{6}}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator \(1 - \sqrt{6}\), we multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 + \sqrt{6}\).
\[
\frac{7}{1 - \sqrt{6}} \times \frac{1 + \sqrt{6}}{1 + \sqrt{6}}
\]

Step2: Multiply the numerators and denominators

Multiply the numerators: \(7\times(1 + \sqrt{6}) = 7 + 7\sqrt{6}\)
Multiply the denominators using the difference of squares formula \((a - b)(a + b)=a^{2}-b^{2}\), here \(a = 1\) and \(b=\sqrt{6}\), so \((1 - \sqrt{6})(1 + \sqrt{6})=1^{2}-(\sqrt{6})^{2}=1 - 6=- 5\)
So we have \(\frac{7 + 7\sqrt{6}}{-5}\)

Step3: Simplify the fraction

We can factor out a negative sign from the denominator and rewrite the fraction as \(-\frac{7 + 7\sqrt{6}}{5}\) or distribute the negative sign in the numerator to get \(\frac{-7 - 7\sqrt{6}}{5}\)

Answer:

\(\boxed{-\dfrac{7 + 7\sqrt{6}}{5}}\) (or \(\boxed{\dfrac{-7 - 7\sqrt{6}}{5}}\))