The problem requires us to simplify the expression $\sqrt{1 \frac{9}{16}}$. This involves taking the square root of a mixed number.

1. Problem Description

The problem requires us to simplify the expression $\sqrt{1 \frac{9}{16}}$. This involves taking the square root of a mixed number.

2. Solution Steps

First, we convert the mixed number $1 \frac{9}{16}$ into an improper fraction.

$1 \frac{9}{16} = \frac{1 \times 16 + 9}{16} = \frac{16 + 9}{16} = \frac{25}{16}$

Now, we can take the square root of the improper fraction:

$\sqrt{1 \frac{9}{16}} = \sqrt{\frac{25}{16}}$

Since the square root of a fraction is the fraction of the square roots, we have:

$\sqrt{\frac{25}{16}} = \frac{\sqrt{25}}{\sqrt{16}}$

We know that $\sqrt{25} = 5$ and $\sqrt{16} = 4$. Therefore,

$\frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4}$

3. Final Answer

$\frac{5}{4}$