`2^5 - 3^4 + 4^3 - 5^2 + 7^2 = (6)^2 + sqrt?`

Options

A. 9
B. 3
C. `(9)^2`
D. `sqrt3`

Answer

To solve the equation \( 2^5 - 3^4 + 4^3 - 5^2 + 7^2 = 6^2 + \sqrt{x} \), we will follow these steps:

### Step 1: Calculate each term on the left side of the equation.

1. Calculate \( 2^5 \):
\[ 2^5 = 32 \]

2. Calculate \( 3^4 \):
\[ 3^4 = 81 \]

3. Calculate \( 4^3 \):
\[ 4^3 = 64 \]

4. Calculate \( 5^2 \):
\[ 5^2 = 25 \]

5. Calculate \( 7^2 \):
\[ 7^2 = 49 \]

### Step 2: Substitute the calculated values into the equation.

Now, substitute these values back into the left side of the equation:
\[ 32 - 81 + 64 - 25 + 49 \]

### Step 3: Perform the arithmetic operations.

1. Start with \( 32 - 81 \):
\[ 32 - 81 = -49 \]

2. Add \( 64 \):
\[ -49 + 64 = 15 \]

3. Subtract \( 25 \):
\[ 15 - 25 = -10 \]

4. Finally, add \( 49 \):
\[ -10 + 49 = 39 \]

So, the left side of the equation simplifies to \( 39 \).

### Step 4: Calculate the right side of the equation.

Now, calculate the right side:
\[ 6^2 + \sqrt{x} \]
Calculate \( 6^2 \):
\[ 6^2 = 36 \]

So, the right side becomes:
\[ 36 + \sqrt{x} \]

### Step 5: Set the left side equal to the right side.

Now we have:
\[ 39 = 36 + \sqrt{x} \]

### Step 6: Solve for \( \sqrt{x} \).

Subtract \( 36 \) from both sides:
\[ 39 - 36 = \sqrt{x} \]
\[ 3 = \sqrt{x} \]

### Step 7: Square both sides to find \( x \).

Square both sides to eliminate the square root:
\[ 3^2 = x \]
\[ x = 9 \]

### Conclusion

The value of \( x \) (or the question mark) is \( 9 \).

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