Question

simplify.
\sqrt{\frac{7}{125}}

Explanation:

Step1: Rationalize the denominator

To simplify $\sqrt{\frac{7}{125}}$, we first rewrite the denominator as a perfect square times another number. We know that $125 = 25\times5$, and $25$ is a perfect square. So we can rewrite the fraction inside the square root as $\frac{7\times5}{125\times5}$ to make the denominator a perfect square.
$$\sqrt{\frac{7}{125}}=\sqrt{\frac{7\times5}{125\times5}}=\sqrt{\frac{35}{625}}$$

Step2: Simplify the square root

Since $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ (for $a\geq0$ and $b > 0$), we can split the square root of the fraction into the square root of the numerator divided by the square root of the denominator. And $\sqrt{625} = 25$ because $25\times25=625$.
$$\sqrt{\frac{35}{625}}=\frac{\sqrt{35}}{\sqrt{625}}=\frac{\sqrt{35}}{25}$$

Answer:

$\frac{\sqrt{35}}{25}$