`47^(7.5)div47^(3//2)xx47^(-3)=(sqrt(47))^(?)`

Options

A. 3
B. `2(1)/(2)`
C. 6
D. 3.5

Explanation

To solve the equation \( 47^{7.5} \div 47^{3/2} \times 47^{-3} = (\sqrt{47})^{?} \), we can simplify the left-hand side step by step.

### Step 1: Simplify the left-hand side using the properties of exponents
We know that \( a^m \div a^n = a^{m-n} \) and \( a^m \times a^n = a^{m+n} \).

So, we can rewrite the left-hand side as:
\[ 47^{7.5} \div 47^{3/2} = 47^{7.5 - 3/2} \]

### Step 2: Calculate \( 7.5 - 3/2 \)
First, convert \( 3/2 \) to a decimal:
\[ 3/2 = 1.5 \]
Now, subtract:
\[ 7.5 - 1.5 = 6 \]
Thus, we have:
\[ 47^{7.5 - 3/2} = 47^6 \]

### Step 3: Include the multiplication by \( 47^{-3} \)
Now, we multiply by \( 47^{-3} \):
\[ 47^6 \times 47^{-3} = 47^{6 - 3} = 47^3 \]

### Step 4: Rewrite the right-hand side
The right-hand side is \( (\sqrt{47})^{?} \). We know that \( \sqrt{47} = 47^{1/2} \), so we can rewrite it as:
\[ (\sqrt{47})^{?} = (47^{1/2})^{?} = 47^{? / 2} \]

### Step 5: Set the exponents equal to each other
Now we have:
\[ 47^3 = 47^{? / 2} \]
Since the bases are the same, we can set the exponents equal:
\[ 3 = \frac{?}{2} \]

### Step 6: Solve for ?
To find \( ? \), multiply both sides by 2:
\[ ? = 3 \times 2 = 6 \]

### Final Answer
Thus, the value of \( ? \) is:
\[ \boxed{6} \]