If `h(x)=2^(kx-1)`, what is the value of `(h(a)h(b))/(h(a+b))` ?

Options

A. 4
B. 2
C. 1
D. 0.5

Answer

To solve the problem, we need to find the value of \(\frac{h(a)h(b)}{h(a+b)}\) given the function \(h(x) = 2^{kx - 1}\).

### Step-by-Step Solution:

1. **Define \(h(a)\)**:
\[ h(a) = 2^{ka - 1} \]
Here, we substitute \(x\) with \(a\) in the function definition.

2. **Define \(h(b)\)**:
\[ h(b) = 2^{kb - 1} \]
Similarly, we substitute \(x\) with \(b\) in the function definition.

3. **Define \(h(a+b)\)**:
\[ h(a+b) = 2^{k(a+b) - 1} \]
Here, we substitute \(x\) with \(a + b\) in the function definition.

4. **Calculate \(h(a)h(b)\)**:
\[ h(a)h(b) = (2^{ka - 1})(2^{kb - 1}) = 2^{(ka - 1) + (kb - 1)} = 2^{ka + kb - 2} \]
We add the exponents since the bases are the same.

5. **Substitute \(h(a+b)\) into the expression**:
\[ \frac{h(a)h(b)}{h(a+b)} = \frac{2^{ka + kb - 2}}{2^{k(a+b) - 1}} \]

6. **Simplify the expression**:
\[ = 2^{(ka + kb - 2) - (k(a+b) - 1)} = 2^{(ka + kb - 2) - (ka + kb - k)} = 2^{-2 + k} \]
Here, we simplify the exponent by distributing the negative sign and combining like terms.

7. **Final simplification**:
\[ = 2^{k - 2} \]

8. **Evaluate the expression**:
Since we need to find the numerical value, we can express \(2^{k - 2}\) as:
\[ = \frac{2^k}{2^2} = \frac{2^k}{4} \]
Without a specific value for \(k\), we cannot simplify further, but we can evaluate it for specific cases if needed.

### Conclusion:
The final expression for \(\frac{h(a)h(b)}{h(a+b)}\) simplifies to \(2^{k - 2}\).