`32*4^?=2048`

Options

A. 3
B. 7
C. 1
D. 2

Answer

To solve the equation \(32 \times 4^? = 2048\), we can follow these steps:

### Step 1: Express the numbers as powers of 2
First, we need to express 32, 4, and 2048 in terms of powers of 2.

- \(32 = 2^5\)
- \(4 = 2^2\)
- \(2048 = 2^{11}\)

### Step 2: Rewrite the equation
Now we can rewrite the original equation using these expressions:
\[ 2^5 \times (2^2)^? = 2^{11} \]

### Step 3: Simplify the equation
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify the left side:
\[ 2^5 \times 2^{2?} = 2^{11} \]

### Step 4: Combine the exponents
Since the bases are the same, we can add the exponents:
\[ 2^{5 + 2?} = 2^{11} \]

### Step 5: Set the exponents equal to each other
Now, we can set the exponents equal to each other:
\[ 5 + 2? = 11 \]

### Step 6: Solve for ?
Subtract 5 from both sides:
\[ 2? = 11 - 5 \]
\[ 2? = 6 \]

Now, divide both sides by 2:
\[ ? = \frac{6}{2} = 3 \]

### Conclusion
The value of the question mark is \(3\).