1. State the problem: Simplify each expression. 2. The three expressions to simplify are: a) $ (5x^2)^3 $. 3. b) $ (16x^4)^{1/4} $. 4. c) $ (64x^9)^{1/3} $. 5. Formula and rules used: the power rules are $$ (ab)^n = a^n b^n $$ and $$ (a^m)^n = a^{mn} $$ These allow us to raise each factor to the outer exponent. 6. Important rule about even roots: when an even root is taken as a principal root, the result is nonnegative, so $ (x^4)^{1/4} = |x| $ for real $x$. 7. Work for (a): Apply the rule $ (a^m)^n = a^{mn} $ to each factor. 8. $ (5x^2)^3 = 5^3 \cdot (x^2)^3 $. 9. $ 5^3 = 125 $ and $ (x^2)^3 = x^{2\cdot 3} = x^6 $. 10. Therefore $ (5x^2)^3 = 125x^6 $. 11. Work for (b): Distribute the exponent $1/4$ to each factor. 12. $ (16x^4)^{1/4} = 16^{1/4} \cdot (x^4)^{1/4} $. 13. $ 16^{1/4} = 2 $ because $2^4 = 16$. 14. For the $x$ part, $ (x^4)^{1/4} = |x| $ as explained above because the fourth root is an even principal root. 15. Therefore the principal real value is $ (16x^4)^{1/4} = 2|x| $. 16. (If you assume $x\ge 0$ you can write the simplified form as $2x$.) 17. Work for (c): Distribute the exponent $1/3$ to each factor. 18. $ (64x^9)^{1/3} = 64^{1/3} \cdot (x^9)^{1/3} $. 19. $ 64^{1/3} = 4 $ because $4^3 = 64$. 20. $ (x^9)^{1/3} = x^{9\cdot(1/3)} = x^3 $ and no absolute value is required because cube roots are odd-indexed. 21. Therefore $ (64x^9)^{1/3} = 4x^3 $. 22. Final answers: 23. a) $125x^6$. 24. b) $2|x|$ (or $2x$ if $x\ge 0$). 25. c) $4x^3$.