`sqrt(33124)xxsqrt(2601)-(83)^(2)=(?)^(2)+(37)^(2)`

Options

A. 37
B. 33
C. 34
D. None of these

Explanation

To solve the equation \( \sqrt{33124} \times \sqrt{2601} - 83^2 = (?)^2 + 37^2 \), we will follow these steps:

### Step 1: Calculate \( \sqrt{33124} \)
We start by finding the square root of 33124.
\[ \sqrt{33124} = 182 \]
**Hint:** To find the square root, you can use a calculator or factor the number into its prime factors.

### Step 2: Calculate \( \sqrt{2601} \)
Next, we find the square root of 2601.
\[ \sqrt{2601} = 51 \]
**Hint:** Again, you can use a calculator or check perfect squares to find this value.

### Step 3: Multiply the square roots
Now, we multiply the results from Step 1 and Step 2.
\[ \sqrt{33124} \times \sqrt{2601} = 182 \times 51 \]
**Hint:** You can use the distributive property or direct multiplication to compute this.

### Step 4: Calculate \( 182 \times 51 \)
Calculating the product:
\[ 182 \times 51 = 9282 \]
**Hint:** Break it down as \( 182 \times (50 + 1) = 182 \times 50 + 182 \times 1 \).

### Step 5: Calculate \( 83^2 \)
Next, we calculate \( 83^2 \).
\[ 83^2 = 6889 \]
**Hint:** You can use the formula \( a^2 = (a)(a) \) or simply multiply 83 by itself.

### Step 6: Subtract \( 83^2 \) from the product
Now we subtract \( 6889 \) from \( 9282 \):
\[ 9282 - 6889 = 2393 \]
**Hint:** Align the numbers vertically to make subtraction easier.

### Step 7: Calculate \( 37^2 \)
Next, we calculate \( 37^2 \).
\[ 37^2 = 1369 \]
**Hint:** Use the same method as before for squaring a number.

### Step 8: Set up the equation
Now we set up the equation:
\[ 2393 = x^2 + 1369 \]
**Hint:** This equation shows that \( x^2 \) is the remaining part after accounting for \( 1369 \).

### Step 9: Solve for \( x^2 \)
We isolate \( x^2 \):
\[ x^2 = 2393 - 1369 \]
Calculating this gives:
\[ x^2 = 1024 \]
**Hint:** Ensure you perform the subtraction correctly.

### Step 10: Find \( x \)
Finally, we take the square root of \( 1024 \):
\[ x = \sqrt{1024} = 32 \]
**Hint:** Recognize that \( 1024 \) is a perfect square, specifically \( 32 \times 32 \).

### Conclusion
The value of \( ? \) is:
\[ ? = 32 \]