Question

simplify. rationalize the denominator. \\(\frac{\sqrt{7}}{-6 - \sqrt{7}}\\)

Explanation:

Step1: Identify conjugate of denominator

The denominator is \(-6 - \sqrt{7}\), its conjugate is \(-6 + \sqrt{7}\) (we change the sign of the radical term).

Step2: Multiply numerator and denominator by conjugate

Multiply \(\frac{\sqrt{7}}{-6 - \sqrt{7}}\) by \(\frac{-6 + \sqrt{7}}{-6 + \sqrt{7}}\):
\[
\frac{\sqrt{7}(-6 + \sqrt{7})}{(-6 - \sqrt{7})(-6 + \sqrt{7})}
\]

Step3: Expand numerator and denominator

• Numerator: Use distributive property (\(a(b + c)=ab + ac\)):

\(\sqrt{7}(-6)+\sqrt{7}\times\sqrt{7}=-6\sqrt{7}+7\)

• Denominator: Use difference of squares formula \((a - b)(a + b)=a^{2}-b^{2}\), here \(a=-6\), \(b = \sqrt{7}\):

\((-6)^{2}-(\sqrt{7})^{2}=36 - 7 = 29\)
So the expression becomes:
\[
\frac{-6\sqrt{7}+7}{29}
\]
We can rewrite the numerator as \(7 - 6\sqrt{7}\), so the simplified form is \(\frac{7 - 6\sqrt{7}}{29}\) (or we can factor out a negative sign from numerator and denominator but this is also a simplified form).

Answer:

\(\frac{7 - 6\sqrt{7}}{29}\) (or \(\frac{-6\sqrt{7}+7}{29}\))