Precalculus Examples

${(\frac{a^{2} b^{- 3}}{x^{- 1} y^{2}})}^{3} (\frac{x^{- 2} b^{- 1}}{a^{\frac{3}{2}} y^{\frac{1}{3}}})$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{a^{2} b^{- 3}}{x^{- 1} y^{2}})}^{3} \frac{\frac{1}{x^{2}} b^{- 1}}{a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{a^{2} b^{- 3}}{x^{- 1} y^{2}})}^{3} \frac{\frac{1}{x^{2}} \cdot \frac{1}{b}}{a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1}{x^{2}}$ by $\frac{1}{b}$ .

${(\frac{a^{2} b^{- 3}}{x^{- 1} y^{2}})}^{3} \frac{\frac{1}{x^{2} b}}{a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $b^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{a^{2}}{x^{- 1} y^{2} b^{3}})}^{3} \frac{\frac{1}{x^{2} b}}{a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 2.2.2

Move $x^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{a^{2} x}{y^{2} b^{3}})}^{3} \frac{\frac{1}{x^{2} b}}{a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $\frac{a^{2} x}{y^{2} b^{3}}$ .

$\frac{{(a^{2} x)}^{3}}{{(y^{2} b^{3})}^{3}} \cdot \frac{\frac{1}{x^{2} b}}{a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 3.2

Apply the product rule to $a^{2} x$ .

$\frac{{(a^{2})}^{3} x^{3}}{{(y^{2} b^{3})}^{3}} \cdot \frac{\frac{1}{x^{2} b}}{a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 3.3

Apply the product rule to $y^{2} b^{3}$ .

$\frac{{(a^{2})}^{3} x^{3}}{{(y^{2})}^{3} {(b^{3})}^{3}} \cdot \frac{\frac{1}{x^{2} b}}{a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 4

Combine.

$\frac{{(a^{2})}^{3} x^{3} \frac{1}{x^{2} b}}{{(y^{2})}^{3} {(b^{3})}^{3} (a^{\frac{3}{2}} y^{\frac{1}{3}})}$

Step 5

Simplify the numerator.

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Step 5.1

Combine ${(a^{2})}^{3}$ and $\frac{1}{x^{2} b}$ .

$\frac{x^{3} \frac{{(a^{2})}^{3}}{x^{2} b}}{{(y^{2})}^{3} {(b^{3})}^{3} (a^{\frac{3}{2}} y^{\frac{1}{3}})}$

Step 5.2

Combine $x^{3}$ and $\frac{{(a^{2})}^{3}}{x^{2} b}$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{{(y^{2})}^{3} {(b^{3})}^{3} (a^{\frac{3}{2}} y^{\frac{1}{3}})}$

Step 6

Simplify the denominator.

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Step 6.1

Remove unnecessary parentheses.

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{{(y^{2})}^{3} {(b^{3})}^{3} a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 6.2

Multiply the exponents in ${(y^{2})}^{3}$ .

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Step 6.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{2 \cdot 3} {(b^{3})}^{3} a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 6.2.2

Multiply $2$ by $3$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{6} {(b^{3})}^{3} a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 6.3

Multiply the exponents in ${(b^{3})}^{3}$ .

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Step 6.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{6} b^{3 \cdot 3} a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 6.3.2

Multiply $3$ by $3$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{6} b^{9} a^{\frac{3}{2}} y^{\frac{1}{3}}}$

Step 6.4

Multiply $y^{6}$ by $y^{\frac{1}{3}}$ by adding the exponents.

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Step 6.4.1

Move $y^{\frac{1}{3}}$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{\frac{1}{3}} y^{6} b^{9} a^{\frac{3}{2}}}$

Step 6.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{\frac{1}{3} + 6} b^{9} a^{\frac{3}{2}}}$

Step 6.4.3

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{\frac{1}{3} + 6 \cdot \frac{3}{3}} b^{9} a^{\frac{3}{2}}}$

Step 6.4.4

Combine $6$ and $\frac{3}{3}$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{\frac{1}{3} + \frac{6 \cdot 3}{3}} b^{9} a^{\frac{3}{2}}}$

Step 6.4.5

Combine the numerators over the common denominator.

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{\frac{1 + 6 \cdot 3}{3}} b^{9} a^{\frac{3}{2}}}$

Step 6.4.6

Simplify the numerator.

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Step 6.4.6.1

Multiply $6$ by $3$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{\frac{1 + 18}{3}} b^{9} a^{\frac{3}{2}}}$

Step 6.4.6.2

Add $1$ and $18$ .

$\frac{\frac{x^{3} {(a^{2})}^{3}}{x^{2} b}}{y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}}}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Multiply the exponents in ${(a^{2})}^{3}$ .

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Step 7.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{x^{3} a^{2 \cdot 3}}{x^{2} b}}{y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}}}$

Step 7.1.2

Multiply $2$ by $3$ .

$\frac{\frac{x^{3} a^{6}}{x^{2} b}}{y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}}}$

Step 7.2

Reduce the expression $\frac{x^{3} a^{6}}{x^{2} b}$ by cancelling the common factors.

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Step 7.2.1

Factor $x^{2}$ out of $x^{3} a^{6}$ .

$\frac{\frac{x^{2} (x a^{6})}{x^{2} b}}{y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}}}$

Step 7.2.2

Factor $x^{2}$ out of $x^{2} b$ .

$\frac{\frac{x^{2} (x a^{6})}{x^{2} (b)}}{y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}}}$

Step 7.2.3

Cancel the common factor.

$\frac{\frac{x^{2} (x a^{6})}{x^{2} b}}{y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}}}$

Step 7.2.4

Rewrite the expression.

$\frac{\frac{x a^{6}}{b}}{y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{x a^{6}}{b} \cdot \frac{1}{y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}}}$

Step 9

Combine.

$\frac{x a^{6} \cdot 1}{b (y^{\frac{19}{3}} b^{9} a^{\frac{3}{2}})}$

Step 10

Multiply $b$ by $b^{9}$ by adding the exponents.

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Step 10.1

Move $b^{9}$ .

$\frac{x a^{6} \cdot 1}{b^{9} b (y^{\frac{19}{3}} a^{\frac{3}{2}})}$

Step 10.2

Multiply $b^{9}$ by $b$ .

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Step 10.2.1

Raise $b$ to the power of $1$ .

$\frac{x a^{6} \cdot 1}{b^{9} b^{1} (y^{\frac{19}{3}} a^{\frac{3}{2}})}$

Step 10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x a^{6} \cdot 1}{b^{9 + 1} (y^{\frac{19}{3}} a^{\frac{3}{2}})}$

Step 10.3

Add $9$ and $1$ .

$\frac{x a^{6} \cdot 1}{b^{10} (y^{\frac{19}{3}} a^{\frac{3}{2}})}$

Step 11

Multiply $x a^{6}$ by $1$ .

$\frac{x a^{6}}{b^{10} (y^{\frac{19}{3}} a^{\frac{3}{2}})}$

Step 12

Move $a^{\frac{3}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{x a^{6} a^{- \frac{3}{2}}}{b^{10} y^{\frac{19}{3}}}$

Step 13

Multiply $a^{6}$ by $a^{- \frac{3}{2}}$ by adding the exponents.

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Step 13.1

Move $a^{- \frac{3}{2}}$ .

$\frac{x (a^{- \frac{3}{2}} a^{6})}{b^{10} y^{\frac{19}{3}}}$

Step 13.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x a^{- \frac{3}{2} + 6}}{b^{10} y^{\frac{19}{3}}}$

Step 13.3

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x a^{- \frac{3}{2} + 6 \cdot \frac{2}{2}}}{b^{10} y^{\frac{19}{3}}}$

Step 13.4

Combine $6$ and $\frac{2}{2}$ .

$\frac{x a^{- \frac{3}{2} + \frac{6 \cdot 2}{2}}}{b^{10} y^{\frac{19}{3}}}$

Step 13.5

Combine the numerators over the common denominator.

$\frac{x a^{\frac{- 3 + 6 \cdot 2}{2}}}{b^{10} y^{\frac{19}{3}}}$

Step 13.6

Simplify the numerator.

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Step 13.6.1

Multiply $6$ by $2$ .

$\frac{x a^{\frac{- 3 + 12}{2}}}{b^{10} y^{\frac{19}{3}}}$

Step 13.6.2

Add $- 3$ and $12$ .

$\frac{x a^{\frac{9}{2}}}{b^{10} y^{\frac{19}{3}}}$