Let's solve each expression step-by-step.

Part (a)

The expression is: $6^{-\frac{1}{4}} \cdot 36^{-\frac{3}{8}}$

1. Evaluate $6^{-\frac{1}{4}}$: $6^{-\frac{1}{4}} = \frac{1}{6^{\frac{1}{4}}} = \frac{1}{\sqrt[4]{6}} \approx 0.4573$

2. Evaluate $36^{-\frac{3}{8}}$: $36^{-\frac{3}{8}} = \frac{1}{36^{\frac{3}{8}}} = \frac{1}{\sqrt[8]{36^3}}$ Since $36 = 6^2$, we can rewrite this as: $36^{-\frac{3}{8}} = (6^2)^{-\frac{3}{8}} = 6^{-\frac{3}{4}} = \frac{1}{6^{\frac{3}{4}}} \approx 0.366$

3. Multiply the results: $6^{-\frac{1}{4}} \cdot 36^{-\frac{3}{8}} \approx 0.4573 \cdot 0.366 \approx 0.167$

Therefore, the answer to (a) is approximately 0.167, which appears correct.

Part (b)

The expression is: $\left(\frac{27}{8}\right)^{-\frac{1}{3}} \cdot \left(\frac{4}{16}\right)^{-\frac{1}{2}}$

1. Evaluate $\left(\frac{27}{8}\right)^{-\frac{1}{3}}$: $\left(\frac{27}{8}\right)^{-\frac{1}{3}} = \frac{1}{\left(\frac{27}{8}\right)^{\frac{1}{3}}} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \approx 0.6667$

2. Evaluate $\left(\frac{4}{16}\right)^{-\frac{1}{2}}$: Simplify $\frac{4}{16}$ first: $\frac{4}{16} = \frac{1}{4}$ Then: $\left(\frac{1}{4}\right)^{-\frac{1}{2}} = \frac{1}{\left(\frac{1}{4}\right)^{\frac{1}{2}}} = \frac{1}{\frac{1}{2}} = 2$

3. Multiply the results: $\left(\frac{27}{8}\right)^{-\frac{1}{3}} \cdot \left(\frac{4}{16}\right)^{-\frac{1}{2}} = 0.6667 \cdot 2 = 1.333$

Thus, the answer to (b) is 1.333, which appears correct.

Would you like further details or have any questions?

Related Questions

1. How would the answers change if we modified the exponents?
2. What is the impact of negative exponents on fractions?
3. How can we generalize these steps to other base numbers?
4. How do we convert fractional exponents to radicals?
5. What are common mistakes to avoid when simplifying expressions with fractional exponents?

Tip

When working with fractional exponents, remember that a negative exponent indicates a reciprocal, and fractional values relate to roots.