Question

Simplify. Rationalize the denominator.
4/–3–√2
4 is numerater
-3-√2 is denominater

Answer 1

To simplify the expression \( \frac{4}{-3 - \sqrt{2}} \) and rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \( -3 - \sqrt{2} \) is \( -3 + \sqrt{2} \).

Here's how to proceed:

1. **Multiply numerator and denominator by the conjugate:**

\[
\frac{4}{-3 - \sqrt{2}} \cdot \frac{-3 + \sqrt{2}}{-3 + \sqrt{2}} = \frac{4(-3 + \sqrt{2})}{(-3 - \sqrt{2})(-3 + \sqrt{2})}
\]

2. **Calculate the denominator using the difference of squares:**

\[
(-3 - \sqrt{2})(-3 + \sqrt{2}) = (-3)^2 - (\sqrt{2})^2 = 9 - 2 = 7
\]

3. **Calculate the numerator:**

\[
4(-3 + \sqrt{2}) = -12 + 4\sqrt{2}
\]

4. **Combine results:**

Putting the simplified numerator and denominator together gives us:

\[
\frac{-12 + 4\sqrt{2}}{7}
\]

Thus, the simplified expression is:

\[
\frac{4\sqrt{2} - 12}{7}
\]

This is the final answer.