Answer
a) $\frac{xy^{3}}{3}$
b) $\frac{y^{6}}{4x^{6}}$
c) $\frac{x^{4}y^{5}}{9}$
Work Step by Step
a) We simplify as follows:
$\frac{3x^{-2}y^{5}}{9x^{-3}y^{2}}$
$=\frac{x^{-2}y^{5}}{3x^{-3}y^{2}}$ Divide the numerator and denominator by 3.
$=\frac{xy^{3}}{3}$ Using exponential laws, simplify $\frac{x^{-2}}{x^{-3}}$ to $x$ and $\frac{y^{5}}{y^{2}}$ to $y^{3}$.
b) We simplify as follows:
$(\frac{2x^{3}y^{-1}}{y^{2}})^{-2}$
$=(2x^{3}y^{-3})^{-2}$ Use exponent laws to simplify $\frac{y^{-1}}{y^{2}}$ to $y^{-3}$.
$=(\frac{1}{2x^{3}y^{-3}})^{2}$ To remove the negative exponent, divide 1 by it to create a reciprocal.
$=(\frac{y^{3}}{2x^{3}})^{2}$ Move $y^{-3}$ to the numerator to get rid of the negative exponent.
$=\frac{y^{6}}{4x^{6}}$ Square everything.
c) We simplify as follows:
$(\frac{y^{-1}}{x^{-2}})^{-1}(\frac{3x^{-3}}{y^{2}})^{-2}$
$=(\frac{x^{-2}}{y^{-1}})(\frac{y^{4}}{9x^{-6}})$ Expand the exponents into the fractions using exponent laws.
$=(\frac{y}{x^{2}})(\frac{y^{4}}{9x^{-6}})$ Remove the negative exponents by creating a reciprocal of the original fraction.
$=\frac{y\times y^{4}}{x^{2}\times 9x^{-6}}$ Multiply the fractions together.
$=\frac{y^{5}}{9x^{-4}}$ Use exponent laws to simplify.
$=\frac{x^{4}y^{5}}{9}$ Remove the negative exponent on $x^{-4}$ by bringing it to the numerator.
