`(sqrt(65025))^(2)=(?)^(2)`

Options

A. `65.025`
B. `32512.5`
C. 255
D. 510

Answer

To solve the equation \((\sqrt{65025})^{2} = (?)^{2}\), we can follow these steps:

### Step 1: Simplify the Left Side
The expression \((\sqrt{65025})^{2}\) can be simplified. The square root and the square cancel each other out.

\[ (\sqrt{65025})^{2} = 65025 \]

### Step 2: Set Up the Equation
Now we have:

\[ 65025 = (?)^{2} \]

### Step 3: Take the Square Root
To find the value of \( ? \), we need to take the square root of both sides of the equation:

\[ ? = \sqrt{65025} \]

### Step 4: Calculate the Square Root
To find \(\sqrt{65025}\), we can factor \(65025\) to find its prime factors.

1. **Finding Prime Factors:**
- \(65025\) is odd, so we can start dividing by \(5\):
- \(65025 \div 5 = 13005\)
- \(13005 \div 5 = 2601\)
- Now, \(2601\) can be divided by \(3\):
- \(2601 \div 3 = 867\)
- \(867 \div 3 = 289\)
- Finally, \(289\) is \(17 \times 17\) (since \(17^2 = 289\)).

Thus, the prime factorization of \(65025\) is:
\[ 65025 = 5^2 \times 3^2 \times 17^2 \]

2. **Taking the Square Root:**
- Now, we can take the square root of the prime factorization:
\[ \sqrt{65025} = \sqrt{(5^2) \times (3^2) \times (17^2)} = 5 \times 3 \times 17 \]
- Calculating this gives:
\[ 5 \times 3 = 15 \]
\[ 15 \times 17 = 255 \]

### Step 5: Conclusion
Thus, we find that:

\[ ? = 255 \]

So the final answer is:

\[ (\sqrt{65025})^{2} = 255^{2} \]