Question

Simplify. Rationalize the denominator.
–7/
10+√2
-7 is numinator
10+√2 is denominer

Answer 1

To simplify the expression \(-\frac{7}{10 + \sqrt{2}}\) and rationalize the denominator, we will multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(10 + \sqrt{2}\) is \(10 - \sqrt{2}\).

Here's the step-by-step process:

1. Multiply the numerator and denominator by the conjugate:
\[
-\frac{7}{10 + \sqrt{2}} \cdot \frac{10 - \sqrt{2}}{10 - \sqrt{2}} = \frac{-7(10 - \sqrt{2})}{(10 + \sqrt{2})(10 - \sqrt{2})}
\]

2. Simplify the denominator using the difference of squares:
\[
(10 + \sqrt{2})(10 - \sqrt{2}) = 10^2 - (\sqrt{2})^2 = 100 - 2 = 98
\]

3. Simplify the numerator:
\[
-7(10 - \sqrt{2}) = -70 + 7\sqrt{2}
\]

4. Combine the results:
\[
\frac{-70 + 7\sqrt{2}}{98}
\]

5. To express it in its simplest form, we can reduce the fraction:
\[
\frac{-70}{98} + \frac{7\sqrt{2}}{98} = -\frac{35}{49} + \frac{\sqrt{2}}{14}
\]

Thus, the simplified expression after rationalizing the denominator is:

\[
-\frac{35}{49} + \frac{\sqrt{2}}{14}
\]
or, combining the parts into one fraction, you could also write it as:

\[
\frac{-35 + \sqrt{2} \cdot 7}{98}
\]

But it's perfectly acceptable to leave it as

\[
-\frac{35}{49} + \frac{\sqrt{2}}{14}
\]

Final answer:
\[
-\frac{35}{49} + \frac{\sqrt{2}}{14}
\] or \(\frac{-70 + 7\sqrt{2}}{98}\)