Question

Simplify the radical expression. $$\sqrt{\frac{20}{25}}$$

Short answer

The simplified form of the expression \(\sqrt{\frac{20}{25}}\) is \(\frac{2\sqrt{5}}{5}\) .

Step-by-step solution

Rewrite the fractions under the radical

Rewrite the ratio \(\frac{20}{25}\) as \(\frac{4}{5} * \frac{5}{5}\). Both of these fractions simplify as well, becoming \(\sqrt{\frac{4}{5}} * \sqrt{\frac{5}{5}}\) which can be further simplified to \(\sqrt{\frac{4}{5}} * 1 \).

Simplify the square root of the fraction

Apply the property of radicals that states the square root of a ratio equals the ratio of the square roots. We get \(\frac{\sqrt{4}}{\sqrt{5}} * 1\). The square root of 4 is 2, so the expression simplifies to \(\frac{2}{\sqrt{5}}\).

Rationalize the denominator

We can't leave a radical in the denominator, so we rationalize the denominator by multiplying the fraction by \(\frac{\sqrt{5}}{\sqrt{5}}\), which is equivalent to 1. This will clear the square root in the denominator. We get \(\frac{2\sqrt{5}}{5}\).