We need to evaluate the square root of the mixed number $75 \frac{46}{49}$.

1. Problem Description

We need to evaluate the square root of the mixed number $75 \frac{46}{49}$.

2. Solution Steps

First, we convert the mixed number to an improper fraction.

$75 \frac{46}{49} = \frac{75 \times 49 + 46}{49} = \frac{3675 + 46}{49} = \frac{3721}{49}$

Then, we find the square root of the fraction.

$\sqrt{75 \frac{46}{49}} = \sqrt{\frac{3721}{49}}$

We know that $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, so we can rewrite the expression as:

$\sqrt{\frac{3721}{49}} = \frac{\sqrt{3721}}{\sqrt{49}}$

We know that $\sqrt{49} = 7$.

We also know that $\sqrt{3721} = 61$, since $61 \times 61 = 3721$.

Therefore,

$\frac{\sqrt{3721}}{\sqrt{49}} = \frac{61}{7}$

Now, we convert the improper fraction to a mixed number.

$\frac{61}{7} = 8 \frac{5}{7}$

So, the square root of $75 \frac{46}{49}$ is $8 \frac{5}{7}$.

3. Final Answer

$8 \frac{5}{7}$