Question

1. ((4x^{\frac{7}{3}})^{-3})\\15. ((8x^{6})^{\frac{4}{3}})

Explanation:

Step1: Apply power of a product rule

$(ab)^n = a^n b^n$
For problem 12: $(4x^{\frac{7}{3}})^{-3} = 4^{-3} \cdot (x^{\frac{7}{3}})^{-3}$
For problem 15: $(8x^6)^{\frac{4}{3}} = 8^{\frac{4}{3}} \cdot (x^6)^{\frac{4}{3}}$

Step2: Simplify constant terms

For problem 12: $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$
For problem 15: $8^{\frac{4}{3}} = (8^{\frac{1}{3}})^4 = 2^4 = 16$

Step3: Simplify variable terms

Power rule: $(a^m)^n = a^{m \cdot n}$
For problem 12: $(x^{\frac{7}{3}})^{-3} = x^{\frac{7}{3} \cdot (-3)} = x^{-7} = \frac{1}{x^7}$
For problem 15: $(x^6)^{\frac{4}{3}} = x^{6 \cdot \frac{4}{3}} = x^8$

Step4: Combine results

For problem 12: Multiply constant and variable parts
For problem 15: Multiply constant and variable parts

Answer:

Problem 12: $\frac{1}{64x^7}$
Problem 15: $16x^8$