Question

# Simplify each of the following expressions. Be sure that your answer has no negative or fractional exponents. a*(1/81)^(-1/4)b*x^(-2)y^(-4)c*(2x)^(-2)(16x^2y)^(1/2)

Simplify each of the following expressions. Be sure that your answer has no negative or fractional exponents. $$a\cdot(\frac{1}{81})^{-\frac{1}{4}}b\cdot x^{-2}y^{-4}c\cdot (2x)^{-2}(16x^{2y})^\frac{1}{2}$$

2021-03-05

a)
$$\frac{1}{81}^{-\frac{1}{4}}$$
$$=(\frac{1}{3})^{-\frac{1}{4}}$$
$$=(\frac{1}{3})$$ (product of power property)
$$(\frac{1}{3})^{-1}=3$$
b)
$$x^2 y^-4$$
$$=x^{-2}\cdot y^{-4}$$
$$\frac{1}{x^{2}}\cdot\frac{1}{y^{4}}$$ (negative power property)
c)
$$(2x)^{-2}(16x^{2y})^{\frac{1}{2}}$$
$$=2^{-2}\cdot x^{-2}\cdot2^{\frac{4}{2}}\cdot x^{\frac{2}{2}}\cdot y^{\frac{1}{2}}$$ (product of power property)
$$=2^{(2-2)}\cdot x^{(1-2)}\cdot y^{\frac{1}{2}}$$
$$=1\cdot x^{-1}\cdot y(\frac{1}{2})$$