Question

Five times radical 108 / 2 times radical 125

Answer 1

To simplify the expression \( \frac{5 \sqrt{108}}{2 \sqrt{125}} \), let's first simplify the square roots.

1. **Simplify \( \sqrt{108} \):**
\[
\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3}
\]

2. **Simplify \( \sqrt{125} \):**
\[
\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}
\]

Now, substituting these back into the original expression:

\[
\frac{5 \sqrt{108}}{2 \sqrt{125}} = \frac{5 \cdot 6\sqrt{3}}{2 \cdot 5\sqrt{5}}
\]

Next, we can cancel out the \( 5 \) in the numerator and denominator:

\[
= \frac{6\sqrt{3}}{2\sqrt{5}}
\]

Now simplify the fraction:

\[
= \frac{6}{2} \cdot \frac{\sqrt{3}}{\sqrt{5}} = 3 \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{3\sqrt{3}}{\sqrt{5}}
\]

To rationalize the denominator, multiply numerator and denominator by \( \sqrt{5} \):

\[
= \frac{3\sqrt{3} \cdot \sqrt{5}}{5} = \frac{3\sqrt{15}}{5}
\]

Thus, the simplified form of the original expression is:

\[
\frac{3\sqrt{15}}{5}
\]