Answer: B. $\text{35}$

Explanation:1. Understand the problem: The function $h$ is defined by $h(x) = a^x + b$, where $a$ and $b$ are positive constants. The graph of $y = h(x)$ passes through two points: $(0, 8)$ and $(2, 32)$. We need to find the value of the product $ab$.

2. Use the given points to form a system of equations: Since the graph passes through $(0, 8)$, we can substitute $x=0$ and $h(x)=8$ into the function: $h(0) = a^0 + b$ $8 = 1 + b$ (because $a^0 = 1$ for any positive constant $a$). From this equation, we can solve for $b$: $b = 8 - 1 = 7$.

Since the graph passes through $(2, 32)$, we can substitute $x=2$ and $h(x)=32$ into the function: $h(2) = a^2 + b$ $32 = a^2 + b$.

3. Solve for $a$ and $b$: We found $b=7$. Substitute this value into the second equation: $32 = a^2 + 7$. Subtract $\text{7}$ from both sides: $a^2 = 32 - 7$ $a^2 = 25$. Since $a$ is a positive constant, take the positive square root: $a = \sqrt{25} = 5$.

4. Calculate the value of $ab$: We have $a=5$ and $b=7$. $ab = 5 \times 7 = 35$.

5. Compare with the given options: [A] $\text{32}$ [B] $\text{35}$ [C] $\text{40}$ [D] $\text{64}$ The calculated value $ab = 35$ matches option [B].

6. Conclusion: The value of $ab$ is $\text{35}$.