What will come in place of question mark (?) in the following equation?
`16^(7 cdot 5) div 8^(3cdot5) div 2^(7 cdot 5) = ?`

Options

A. `8^4`
B. `16^4`
C. `2^15`
D. `2^27`

Answer

To solve the equation \( 16^{(7 \cdot 5)} \div 8^{(3 \cdot 5)} \div 2^{(7 \cdot 5)} = ? \), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
We know that:
- \( 16 = 2^4 \)
- \( 8 = 2^3 \)

So we can rewrite the equation as:
\[ (2^4)^{(7 \cdot 5)} \div (2^3)^{(3 \cdot 5)} \div 2^{(7 \cdot 5)} \]

### Step 2: Simplify the exponents
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we can simplify the expression:
\[ 2^{(4 \cdot 35)} \div 2^{(3 \cdot 15)} \div 2^{(7 \cdot 5)} \]
Calculating the exponents:
- \( 4 \cdot 35 = 140 \)
- \( 3 \cdot 15 = 45 \)
- \( 7 \cdot 5 = 35 \)

Now we have:
\[ 2^{140} \div 2^{45} \div 2^{35} \]

### Step 3: Combine the divisions
Using the property of exponents \( a^m \div a^n = a^{m-n} \), we can combine the terms:
\[ 2^{140 - 45 - 35} \]

### Step 4: Calculate the exponent
Now we will perform the subtraction:
\[ 140 - 45 = 95 \]
\[ 95 - 35 = 60 \]

So we have:
\[ 2^{60} \]

### Step 5: Final answer
Thus, the value that comes in place of the question mark is:
\[ 2^{60} \]