Question

Let \(f(x) = a^{3x}\) and \(a^5 = 8\). Then the value of \(f(5)\) is equal to:

• A. 64
• B. 128
• C. 256
• D. 512
• E. 1024

Hint:

When dealing with exponents and roots, simplify the expression by finding the base value first, and then apply the exponents as needed. This often simplifies the calculations significantly.

Answer 1

The Correct Option is D

Given the function \(f(x) = a^{3x}\) and the equation \(a^5 = 8\), we first need to find \(a\). 
Since \(a^5 = 8\), we can solve for \(a\) as follows: \[ a = 8^{1/5} \] \[ a = 2^{3/5} \] Now, calculate \(f(5)\): \[ f(5) = a^{3 \times 5} = a^{15} \] Substitute \(a = 2^{3/5}\): \[ a^{15} = (2^{3/5})^{15} = 2^{(3/5) \times 15} = 2^9 = 512 \] Thus, \(f(5) = 512\).