Answer
$\dfrac{18x^5}{y^2}$
Restrictions: $x \ne 0$, $y \ne 0$
Work Step by Step
Rewrite the exercise using the division ($\div$) symbol:
$\left(54x^3y^{-1}\right) \div \left({3x^{-2}y}\right)$
Move the expression with the negative exponent from the denominator to the numerator and the one in the numerator to the denominator:
$=\left(54x^3 \cdot x^{2}\right) \div \left({3y \cdot y}\right)$
To divide one rational expression by another, multiply by the reciprocal:
$=54x^3 \cdot x^{2} \cdot \dfrac{1}{3y \cdot y}$
Rewrite as one rational expression:
$=\dfrac{54x^3 \cdot x^{2}}{3y \cdot y}$
Cancel out common factors:
$=\dfrac{18x^3 \cdot x^{2}}{y \cdot y}$
Simplify:
$=\dfrac{18x^5}{y^2}$
Restrictions occur when the expression becomes undefined, meaning when the denominator is $0$.
To find the restrictions, set each expression in the denominators of the original rational expressions as well as in the reciprocal equal to zero and solve:
Restriction:
From the denominators, we can see that neither $x$ nor $y$ can be $0$.
Therefore, the restrictions are $x \ne 0$, $y \ne 0$.
