Answer
a. $\dfrac{1}{8}$
b. $8$
c. $\dfrac{1}{2187}$
Work Step by Step
a. Negative exponents means we move the radical expression to the denominator, changing the exponent from negative to positive:
$\dfrac{1}{32^{\frac{3}{5}}}$
Convert exponential expressions to radicals:
$=\dfrac{1}{\sqrt[5] {32^{3}}}$
Factor the radicand:
$=\dfrac{1}{\sqrt[5] {({2^{5})^{3}}}}$
When raising a power to a power, multiply the exponents, keeping the base as-is:
$=\dfrac{1}{\sqrt[5] {2^{15}}}$
Factor such that we can take the fifth root of the radicand:
$=\dfrac{1}{\sqrt[5] {2^{5} \cdot 2^{5} \cdot 2^{5}}}$
Take the fifth roots:
$=\dfrac{1}{2 \cdot 2 \cdot 2}$
Multiply to simplify:
$=\dfrac{1}{8}$
b. Convert radicals to exponential expressions:
$\sqrt[4] {16^3}$
Rewrite the radicand such that we can take the fourth root later on:
$=\sqrt[4] {(2^{4})^{3}}$
When raising a power to a power, multiply the exponents, keeping the base as-is:
$=\sqrt[4] {2^{12}}$
Factor such that we can take the fourth root of the radicand:
$=\sqrt[4] {2^{4} \cdot 2^{4} \cdot 2^{4}}$
Take the fourth roots:
$=2 \cdot 2 \cdot 2$
$=8$
c. Negative exponents means we move the radical expression to the denominator, changing the exponent from negative to positive:
$\dfrac{1}{9^{3.5}}$
Convert the decimal in the exponent into a fraction:
$=\dfrac{1}{9^{\frac{7}{2}}}$
Factor the radicand:
$=\dfrac{1}{\sqrt {(3^{2})^{7}}}$
When raising a power to a power, multiply the exponents, keeping the base as-is:
$=\frac{1}{\sqrt {3^{14}}}$
Rewrite the radicand such that the square roots can be taken:
$=\dfrac{1}{\sqrt {3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2}}$
Take the square roots:
$=\dfrac{1}{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}$
Multiply to simplify:
$=\dfrac{1}{2187}$
