Question

Rewrite the expression in simplest form. Rationalize the denominator.
\sqrt{\frac{8}{7}}

Answer 1

To simplify the expression \(\sqrt{\frac{8}{7}}\) and rationalize the denominator, we can follow these steps:

1. Split the square root:
\[
\sqrt{\frac{8}{7}} = \frac{\sqrt{8}}{\sqrt{7}}
\]

2. Simplify \(\sqrt{8}\):
\[
\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
\]
Therefore, we have:
\[
\frac{\sqrt{8}}{\sqrt{7}} = \frac{2\sqrt{2}}{\sqrt{7}}
\]

3. To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{7}\):
\[
\frac{2\sqrt{2}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{2\sqrt{14}}{7}
\]

Thus, the expression \(\sqrt{\frac{8}{7}}\) in simplest form, with the denominator rationalized, is:
\[
\frac{2\sqrt{14}}{7}
\]