What will come in place of the question mark(?) in the following questions ? (a,b,c are integers)
`(a)^b(a^?)=1`

Options

A. 1
B. c
C. a
D. None of these

Answer

To solve the equation \( a^b \cdot a^{?} = 1 \), we need to find the value of the question mark (?). Let's break it down step by step.

### Step 1: Understand the equation
The equation given is \( a^b \cdot a^{?} = 1 \). We know that when two powers with the same base are multiplied, we can add their exponents.

### Step 2: Apply the exponent rule
Using the exponent rule, we can rewrite the left-hand side:
\[ a^b \cdot a^{?} = a^{b + ?} \]
So, we can rewrite our equation as:
\[ a^{b + ?} = 1 \]

### Step 3: Set the exponent equal to zero
For any base \( a \) (where \( a \neq 0 \)), the only way for \( a^{x} = 1 \) is if the exponent \( x \) is equal to 0. Therefore, we set:
\[ b + ? = 0 \]

### Step 4: Solve for the question mark
Now, we can solve for the question mark (?):
\[ ? = -b \]

### Conclusion
Thus, the value that will come in place of the question mark (?) is:
\[ ? = -b \]