Find the value of question mark (?) in the following questions.
`(?)^((3)/(2)) = 256 xx (2)^(8) div (8)^(5) xx 32`

Options

A. 4
B. 256
C. 64
D. 16

Answer

To find the value of the question mark (?) in the equation

\[ (?)^{\frac{3}{2}} = 256 \times 2^{8} \div 8^{5} \times 32 \]

we will solve the equation step by step.

### Step 1: Rewrite the equation
First, we can rewrite the equation for clarity:

\[ (?)^{\frac{3}{2}} = 256 \times 2^{8} \div 8^{5} \times 32 \]

### Step 2: Simplify the right-hand side
We know that \( 256 = 2^8 \) and \( 8 = 2^3 \), so we can rewrite \( 8^5 \) as \( (2^3)^5 = 2^{15} \). Also, \( 32 = 2^5 \).

Now substituting these values in:

\[ (?)^{\frac{3}{2}} = 2^8 \times 2^{8} \div 2^{15} \times 2^5 \]

### Step 3: Combine the powers of 2
Now, we can simplify the right-hand side:

\[ (?)^{\frac{3}{2}} = \frac{2^8 \times 2^8 \times 2^5}{2^{15}} \]

Using the property of exponents that states \( a^m \times a^n = a^{m+n} \):

\[ (?)^{\frac{3}{2}} = \frac{2^{8+8+5}}{2^{15}} = \frac{2^{21}}{2^{15}} = 2^{21-15} = 2^{6} \]

### Step 4: Solve for the question mark
Now we have:

\[ (?)^{\frac{3}{2}} = 2^{6} \]

To eliminate the exponent \(\frac{3}{2}\), we can raise both sides to the power of \(\frac{2}{3}\):

\[ ? = (2^{6})^{\frac{2}{3}} = 2^{6 \times \frac{2}{3}} = 2^{4} \]

### Step 5: Calculate the final value
Now, we can calculate \(2^4\):

\[ 2^4 = 16 \]

Thus, the value of the question mark (?) is:

\[ \boxed{16} \]