Exercise 21 Page 531

We want to divide the given rational expressions.

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6 y ^{- 3}}{5 x ^{- 5}}

To divide the expressions, let's recall the Negative Exponent Property.

x^{- n} = \frac{1}{x ^{n}}

This property can be rewritten for cases where we have a negative exponent in the denominator.

\frac{1}{x ^{- n}} = x^{n}

Let's use the above to start simplifying the expressions. We will also split the numerators and denominators of both expressions into factors to cancel out any common factors.

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6 y ^{- 3}}{5 x ^{- 5}}

WriteProdFrac

Write as a product of fractions

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6 y ^{- 3}}{5} \cdot \frac{1}{x ^{- 5}}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6}{5} \cdot y^{- 3} \cdot \frac{1}{x ^{- 5}}

NegExponentToFrac

$a^{- m} = \frac{1}{a ^{m}}$

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6}{5} \cdot \frac{1}{y ^{3}} \cdot \frac{1}{x ^{- 5}}

$\frac{1}{x ^{- n}} = x^{n}$

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6}{5} \cdot \frac{1}{y ^{3}} \cdot x^{5}

MultFrac

Multiply fractions

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6}{5 y ^{3}} \cdot x^{5}

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6 x ^{5}}{5 y ^{3}}

WriteSum

Write as a sum

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6 x ^{3 + 2}}{5 y ^{2 + 1}}

SumInExponent

$a^{m + n} = a^{m} \cdot a^{n}$

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6 x ^{3} \cdot x ^{2}}{5 y ^{2} \cdot y ^{1}}

ExponentOne

$a^{1} = a$

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{6 x ^{3} \cdot x ^{2}}{5 y ^{2} \cdot y}

SplitIntoFactors

Split into factors

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{2 \cdot 3 x ^{3} \cdot x ^{2}}{5 y ^{2} \cdot y}

To divide the expressions, we will multiply the first expression by the reciprocal of the second expression.

\frac{3 x ^{3}}{5 y ^{2}} \div \frac{2 \cdot 3 x ^{3} \cdot x ^{2}}{5 y ^{2} \cdot y} ⇕ \frac{3 x ^{3}}{5 y ^{2}} \cdot \frac{5 y ^{2} \cdot y}{2 \cdot 3 x ^{3} \cdot x ^{2}}

Finally, we can cancel out any common factors.

\frac{3 x ^{3}}{5 y ^{2}} \cdot \frac{5 y ^{2} \cdot y}{2 \cdot 3 x ^{3} \cdot x ^{2}}

CancelCommonFac

Cancel out common factors

\frac{3 x ^{3}}{5 y ^{2}} \cdot \frac{5 y ^{2} \cdot y}{2 \cdot 3 x ^{3} \cdot x ^{2}}

SimpQuot

Simplify quotient

\frac{y}{2 \cdot x ^{2}}

Multiply

\frac{y}{2 x ^{2}}

To identify the restrictions on the variables, we need to find any values of

x

and

y

that would cause the denominator of the simplified expression, or any other denominator used, to be

0 .

Denominator	Restrictions on the denominator	Restrictions on the variables
$5 y^{2}$	$5 y^{2} \neq = 0$	$y \neq = 0$
$5 x^{- 5}$	$x^{5} \neq = 0$	$x \neq = 0$
$5 y^{2} \cdot y$	$y^{2} \neq = 0$ and $y \neq = 0$	$y \neq = 0$ and $y \neq = 0$
$2 \cdot 3 x^{3} \cdot x^{2}$	$3 x^{3} \neq = 0$ and $x^{2} \neq = 0$	$x \neq = 0$ and $x \neq = 0$
$2 x^{2}$	$2 x^{2} \neq = 0$	$x \neq = 0$

We found two restrictions on the variables.

y \neq = 0, x \neq = 0