Question

simplify. rationalize the denominator.\frac{-10}{9 + \sqrt{2}}

Explanation:

Step1: Identify conjugate

The conjugate of \(9 + \sqrt{2}\) is \(9 - \sqrt{2}\). Multiply numerator and denominator by this conjugate.
\[
\frac{-10}{9 + \sqrt{2}} \times \frac{9 - \sqrt{2}}{9 - \sqrt{2}}
\]

Step2: Multiply numerators

Multiply the numerators: \(-10 \times (9 - \sqrt{2}) = -90 + 10\sqrt{2}\)

Step3: Multiply denominators

Use the difference of squares formula \((a + b)(a - b)=a^2 - b^2\) for the denominator: \(9^2 - (\sqrt{2})^2 = 81 - 2 = 79\)

Step4: Combine results

Put the numerator and denominator together: \(\frac{-90 + 10\sqrt{2}}{79}\) or \(\frac{10\sqrt{2}-90}{79}\)

Answer:

\(\frac{-90 + 10\sqrt{2}}{79}\) (or \(\frac{10\sqrt{2}-90}{79}\))