Let's simplify each expression step by step:

First Expression:

$\sqrt[4]{(32a^5)^4}$

• The fourth root and exponent 4 cancel each other out: $32a^5$
• So, the simplified answer is: $\mathbf{32a}$

Second Expression:

$\sqrt[8]{\left( b^{\frac{3}{4}} \right)^9}$

• Using the power rule $(a^m)^n = a^{m \cdot n}$: $b^{\left(\frac{3}{4} \times 9\right)}$ $b^{\frac{27}{4}}$
• Applying the 8th root: $b^{\frac{27}{4} \div 8} = b^{\frac{27}{32}}$

Third Expression:

$\sqrt[3]{\frac{c^{14}}{c^8}}$

• Simplify the fraction: $c^{14-8} = c^6$
• Taking the cube root: $c^{6/3} = c^2$

Fourth Expression:

$\sqrt[4]{81d^3 \times 81d^{-6}}$

• Simplify the base $81 = 3^4$, so: $\sqrt[4]{(3^4)d^{3+(-6)}}$ $\sqrt[4]{3^4 d^{-3}}$
• Taking the fourth root: $3^{4/4} d^{-3/4}$ $3d^{-3/4}$

Final Answers:

1. $32a$
2. $b^{\frac{27}{32}}$
3. $c^2$
4. $3d^{-3/4}$

Would you like detailed steps for any specific part?

Here are five related questions to explore further:

1. How do you simplify expressions with radicals and exponents?
2. What are the rules for fractional exponents?
3. How do you simplify nested roots?
4. How do you convert radical expressions into exponent form?
5. How do you handle negative exponents in fraction form?

Tip: When simplifying roots and exponents, always rewrite them as fractions to make calculations easier!