Find the square root of 2304.

You're on the right track with the long division method for $\sqrt{2304}$, but there's a small error in your calculation. Let's re-examine the steps.

Here's the correct way to find the square root of 2304 using the long division method:

Step 1: Pair the digits from the right: $23 \ 04$.

Step 2: Find the largest number whose square is less than or equal to 23. This is 4, since $4^2 = 16$.

Step 3: Write 4 as the first digit of the quotient. Subtract 16 from 23 to get 7. Bring down the next pair, 04, to make the new dividend 704.

Step 4: Double the current quotient (4) to get 8. Now, we need to find a digit 'x' such that $8x \times x \le 704$. * You correctly identified that the next digit in the quotient is 8, making the quotient 48. * So, we calculate $88 \times 8$. * $88 \times 8 = 704$.

Step 5: Write 8 as the next digit in the quotient. Subtract 704 from 704 to get 0.

Your mistake was in the multiplication and subtraction in Step 4. You wrote 64 instead of 704 when multiplying $88 \times 8$.

Here's how the long division should look:

$$\begin{array}{r} 48 \\[-3pt] 4 \overline{\smash{)} 23 \ 04} \\[-3pt] -16 \downarrow \\[-3pt] \hline 88 \quad 704 \\[-3pt] -704 \\[-3pt] \hline 0 \end{array}$$

The square root of 2304 is 48.

$$\sqrt{2304} = \boxed{48}$$