1. We start with the equation given: $$\frac{1}{4 \sqrt{a^3}} = \frac{1}{32}$$ 2. To solve for $a$, first recognize that the denominators must be equal since the fractions are equal. So, set: $$4 \sqrt{a^3} = 32$$ 3. Divide both sides by 4 to isolate the radical: $$\cancel{4} \sqrt{a^3} = \frac{32}{\cancel{4}}$$ $$\sqrt{a^3} = 8$$ 4. Recall that the fourth root means: $$\sqrt{a^3} = (a^3)^{\frac{1}{4}} = a^{\frac{3}{4}}$$ So the equation becomes: $$a^{\frac{3}{4}} = 8$$ 5. To solve for $a$, raise both sides to the power of $\frac{4}{3}$ to cancel the exponent on the left: $$\left(a^{\frac{3}{4}}\right)^{\frac{4}{3}} = 8^{\frac{4}{3}}$$ 6. Simplify the left side by multiplying exponents: $$a^{\frac{3}{4} \times \frac{4}{3}} = a^1 = a$$ 7. Simplify the right side: $$8^{\frac{4}{3}} = \left(8^{\frac{1}{3}}\right)^4 = 2^4 = 16$$ 8. Therefore, the solution is: $$a = 16$$ 9. The function value $f(a)$ is then: $$f(a) = 16$$ Summary: We isolated the radical, converted the root to an exponent, then raised both sides to the reciprocal power to solve for $a$. The final answer is $a=16$.