The square root of 57600 is :

Options

A. 359
B. 323
C. 227
D. 240

Answer

To find the square root of 57600, we will use the method of prime factorization. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 57600
We start by dividing 57600 by the smallest prime number, which is 2.

- 57600 ÷ 2 = 28800
- 28800 ÷ 2 = 14400
- 14400 ÷ 2 = 7200
- 7200 ÷ 2 = 3600
- 3600 ÷ 2 = 1800
- 1800 ÷ 2 = 900
- 900 ÷ 2 = 450
- 450 ÷ 2 = 225

Now, we can no longer divide by 2. Next, we divide by the next smallest prime number, which is 3.

- 225 ÷ 3 = 75
- 75 ÷ 3 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1

### Step 2: Count the Prime Factors
From the divisions, we have:
- 2 appears 8 times (2^8)
- 3 appears 2 times (3^2)
- 5 appears 2 times (5^2)

Thus, we can express 57600 as:
\[ 57600 = 2^8 \times 3^2 \times 5^2 \]

### Step 3: Finding the Square Root
To find the square root, we take the square root of each factor:
\[ \sqrt{57600} = \sqrt{2^8 \times 3^2 \times 5^2} \]

Using the property of square roots:
\[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]
We can separate the square roots:
\[ \sqrt{57600} = \sqrt{2^8} \times \sqrt{3^2} \times \sqrt{5^2} \]

Calculating each square root:
- \( \sqrt{2^8} = 2^{8/2} = 2^4 = 16 \)
- \( \sqrt{3^2} = 3^{2/2} = 3^1 = 3 \)
- \( \sqrt{5^2} = 5^{2/2} = 5^1 = 5 \)

### Step 4: Multiply the Results
Now we multiply the results:
\[ \sqrt{57600} = 16 \times 3 \times 5 \]

Calculating step-by-step:
1. \( 16 \times 3 = 48 \)
2. \( 48 \times 5 = 240 \)

Thus, the square root of 57600 is:
\[ \sqrt{57600} = 240 \]

### Final Answer
The square root of 57600 is **240**.

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