The problem requires us to order the following numbers from least to greatest: $\sqrt{34}$, $5.57$, and $5\frac{3}{4}$.

1. Problem Description

The problem requires us to order the following numbers from least to greatest: $\sqrt{34}$, $5.57$, and $5\frac{3}{4}$.

2. Solution Steps

First, we need to find the approximate value of $\sqrt{34}$. We know that $5^2 = 25$ and $6^2 = 36$. Since 34 is close to 36, $\sqrt{34}$ should be slightly less than

6. A calculator gives $\sqrt{34} \approx 5.83$.

Next, we convert the mixed number $5\frac{3}{4}$ to a decimal.

$5\frac{3}{4} = 5 + \frac{3}{4} = 5 + 0.75 = 5.75$.

Now we have three decimal numbers to compare: $5.83$, $5.57$, and $5.75$. Ordering these from least to greatest gives us: $5.57 < 5.75 < 5.83$.

Therefore, the order from least to greatest is: $5.57$, $5\frac{3}{4}$, $\sqrt{34}$.

3. Final Answer

$5.57, 5\frac{3}{4}, \sqrt{34}$