The problem asks to find the square root of the fraction $\frac{625}{1296}$. That is, we need to evaluate $\sqrt{\frac{625}{1296}}$.

1. Problem Description

The problem asks to find the square root of the fraction $\frac{625}{1296}$. That is, we need to evaluate $\sqrt{\frac{625}{1296}}$.

2. Solution Steps

To find the square root of a fraction, we can take the square root of the numerator and the square root of the denominator separately.

$$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$

So we have:

$$\sqrt{\frac{625}{1296}} = \frac{\sqrt{625}}{\sqrt{1296}}$$

We need to find the square root of 625 and

1

2

9

6. We can find $\sqrt{625}$ by prime factorization.

$625 = 5 \cdot 125 = 5 \cdot 5 \cdot 25 = 5 \cdot 5 \cdot 5 \cdot 5 = 5^4$.

Then $\sqrt{625} = \sqrt{5^4} = 5^2 = 25$.

Now, we need to find $\sqrt{1296}$.

$1296 = 2 \cdot 648 = 2 \cdot 2 \cdot 324 = 2^2 \cdot 324 = 2^2 \cdot 2 \cdot 162 = 2^3 \cdot 162 = 2^3 \cdot 2 \cdot 81 = 2^4 \cdot 81 = 2^4 \cdot 9 \cdot 9 = 2^4 \cdot 9^2 = 2^4 \cdot (3^2)^2 = 2^4 \cdot 3^4 = (2 \cdot 3)^4 = 6^4$.

Thus, $\sqrt{1296} = \sqrt{6^4} = 6^2 = 36$.

Alternatively, we can find $\sqrt{1296}$ by prime factorization:

$1296 = 2 \times 648 = 2^2 \times 324 = 2^3 \times 162 = 2^4 \times 81 = 2^4 \times 3 \times 27 = 2^4 \times 3^2 \times 9 = 2^4 \times 3^4 = (2 \times 3)^4 = 6^4$.

$\sqrt{1296} = \sqrt{2^4 \times 3^4} = \sqrt{(2^2)^2 \times (3^2)^2} = 2^2 \times 3^2 = 4 \times 9 = 36$.

So, we have:

$$\sqrt{\frac{625}{1296}} = \frac{25}{36}$$

3. Final Answer

$\frac{25}{36}$