Precalculus Examples

${(\frac{a^{\frac{1}{12}} b^{- 4}}{x^{- 1} y})}^{4} (\frac{x^{- 3} b^{- 1}}{a^{\frac{4}{3}} y^{\frac{1}{4}}})$

Step 1

Simplify the numerator.

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Step 1.1

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

${(\frac{a^{\frac{1}{12}} b^{- 4}}{x^{- 1} y})}^{4} \frac{\frac{1}{x^{3}} b^{- 1}}{a^{\frac{4}{3}} y^{\frac{1}{4}}}$

Step 1.2

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

${(\frac{a^{\frac{1}{12}} b^{- 4}}{x^{- 1} y})}^{4} \frac{\frac{1}{x^{3}} \cdot \frac{1}{b}}{a^{\frac{4}{3}} y^{\frac{1}{4}}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1}{x^{3}}$ by $\frac{1}{b}$ .

${(\frac{a^{\frac{1}{12}} b^{- 4}}{x^{- 1} y})}^{4} \frac{\frac{1}{x^{3} b}}{a^{\frac{4}{3}} y^{\frac{1}{4}}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $b^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{a^{\frac{1}{12}}}{x^{- 1} y b^{4}})}^{4} \frac{\frac{1}{x^{3} b}}{a^{\frac{4}{3}} y^{\frac{1}{4}}}$

Step 2.2.2

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

${(\frac{a^{\frac{1}{12}} x}{y b^{4}})}^{4} \frac{\frac{1}{x^{3} b}}{a^{\frac{4}{3}} y^{\frac{1}{4}}}$

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 $\frac{a^{\frac{1}{12}} x}{y b^{4}}$ .

$\frac{{(a^{\frac{1}{12}} x)}^{4}}{{(y b^{4})}^{4}} \cdot \frac{\frac{1}{x^{3} b}}{a^{\frac{4}{3}} y^{\frac{1}{4}}}$

Step 3.2

Apply the product rule to $a^{\frac{1}{12}} x$ .

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4}}{{(y b^{4})}^{4}} \cdot \frac{\frac{1}{x^{3} b}}{a^{\frac{4}{3}} y^{\frac{1}{4}}}$

Step 3.3

Apply the product rule to $y b^{4}$ .

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4}}{y^{4} {(b^{4})}^{4}} \cdot \frac{\frac{1}{x^{3} b}}{a^{\frac{4}{3}} y^{\frac{1}{4}}}$

Step 4

Combine.

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4} \frac{1}{x^{3} b}}{y^{4} {(b^{4})}^{4} (a^{\frac{4}{3}} y^{\frac{1}{4}})}$

Step 5

Multiply $y^{4}$ by $y^{\frac{1}{4}}$ by adding the exponents.

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Step 5.1

Move $y^{\frac{1}{4}}$ .

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4} \frac{1}{x^{3} b}}{y^{\frac{1}{4}} y^{4} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 5.2

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

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4} \frac{1}{x^{3} b}}{y^{\frac{1}{4} + 4} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 5.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4} \frac{1}{x^{3} b}}{y^{\frac{1}{4} + 4 \cdot \frac{4}{4}} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 5.4

Combine $4$ and $\frac{4}{4}$ .

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4} \frac{1}{x^{3} b}}{y^{\frac{1}{4} + \frac{4 \cdot 4}{4}} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 5.5

Combine the numerators over the common denominator.

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4} \frac{1}{x^{3} b}}{y^{\frac{1 + 4 \cdot 4}{4}} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $4$ by $4$ .

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4} \frac{1}{x^{3} b}}{y^{\frac{1 + 16}{4}} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 5.6.2

Add $1$ and $16$ .

$\frac{{(a^{\frac{1}{12}})}^{4} x^{4} \frac{1}{x^{3} b}}{y^{\frac{17}{4}} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 6

Simplify the numerator.

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Step 6.1

Combine ${(a^{\frac{1}{12}})}^{4}$ and $\frac{1}{x^{3} b}$ .

$\frac{x^{4} \frac{{(a^{\frac{1}{12}})}^{4}}{x^{3} b}}{y^{\frac{17}{4}} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 6.2

Combine $x^{4}$ and $\frac{{(a^{\frac{1}{12}})}^{4}}{x^{3} b}$ .

$\frac{\frac{x^{4} {(a^{\frac{1}{12}})}^{4}}{x^{3} b}}{y^{\frac{17}{4}} {(b^{4})}^{4} a^{\frac{4}{3}}}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Multiply the exponents in ${(b^{4})}^{4}$ .

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Step 7.1.1

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

$\frac{\frac{x^{4} {(a^{\frac{1}{12}})}^{4}}{x^{3} b}}{y^{\frac{17}{4}} b^{4 \cdot 4} a^{\frac{4}{3}}}$

Step 7.1.2

Multiply $4$ by $4$ .

$\frac{\frac{x^{4} {(a^{\frac{1}{12}})}^{4}}{x^{3} b}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 7.2

Multiply the exponents in ${(a^{\frac{1}{12}})}^{4}$ .

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Step 7.2.1

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

$\frac{\frac{x^{4} a^{\frac{1}{12} \cdot 4}}{x^{3} b}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 7.2.2

Cancel the common factor of $4$ .

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Step 7.2.2.1

Factor $4$ out of $12$ .

$\frac{\frac{x^{4} a^{\frac{1}{4 (3)} \cdot 4}}{x^{3} b}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 7.2.2.2

Cancel the common factor.

$\frac{\frac{x^{4} a^{\frac{1}{4 \cdot 3} \cdot 4}}{x^{3} b}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 7.2.2.3

Rewrite the expression.

$\frac{\frac{x^{4} a^{\frac{1}{3}}}{x^{3} b}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 7.3

Reduce the expression $\frac{x^{4} a^{\frac{1}{3}}}{x^{3} b}$ by cancelling the common factors.

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Step 7.3.1

Factor $x^{3}$ out of $x^{4} a^{\frac{1}{3}}$ .

$\frac{\frac{x^{3} (x a^{\frac{1}{3}})}{x^{3} b}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 7.3.2

Factor $x^{3}$ out of $x^{3} b$ .

$\frac{\frac{x^{3} (x a^{\frac{1}{3}})}{x^{3} (b)}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 7.3.3

Cancel the common factor.

$\frac{\frac{x^{3} (x a^{\frac{1}{3}})}{x^{3} b}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 7.3.4

Rewrite the expression.

$\frac{\frac{x a^{\frac{1}{3}}}{b}}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{x a^{\frac{1}{3}}}{b} \cdot \frac{1}{y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}}}$

Step 9

Combine.

$\frac{x a^{\frac{1}{3}} \cdot 1}{b (y^{\frac{17}{4}} b^{16} a^{\frac{4}{3}})}$

Step 10

Multiply $b$ by $b^{16}$ by adding the exponents.

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Step 10.1

Move $b^{16}$ .

$\frac{x a^{\frac{1}{3}} \cdot 1}{b^{16} b (y^{\frac{17}{4}} a^{\frac{4}{3}})}$

Step 10.2

Multiply $b^{16}$ by $b$ .

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Step 10.2.1

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

$\frac{x a^{\frac{1}{3}} \cdot 1}{b^{16} b^{1} (y^{\frac{17}{4}} a^{\frac{4}{3}})}$

Step 10.2.2

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

$\frac{x a^{\frac{1}{3}} \cdot 1}{b^{16 + 1} (y^{\frac{17}{4}} a^{\frac{4}{3}})}$

Step 10.3

Add $16$ and $1$ .

$\frac{x a^{\frac{1}{3}} \cdot 1}{b^{17} (y^{\frac{17}{4}} a^{\frac{4}{3}})}$

Step 11

Multiply $x a^{\frac{1}{3}}$ by $1$ .

$\frac{x a^{\frac{1}{3}}}{b^{17} (y^{\frac{17}{4}} a^{\frac{4}{3}})}$

Step 12

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

$\frac{x}{b^{17} (y^{\frac{17}{4}} a^{\frac{4}{3}}) a^{- \frac{1}{3}}}$

Step 13

Simplify the denominator.

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Step 13.1

Multiply $a^{\frac{4}{3}}$ by $a^{- \frac{1}{3}}$ by adding the exponents.

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Step 13.1.1

Move $a^{- \frac{1}{3}}$ .

$\frac{x}{b^{17} (y^{\frac{17}{4}} (a^{- \frac{1}{3}} a^{\frac{4}{3}}))}$

Step 13.1.2

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

$\frac{x}{b^{17} (y^{\frac{17}{4}} a^{- \frac{1}{3} + \frac{4}{3}})}$

Step 13.1.3

Combine the numerators over the common denominator.

$\frac{x}{b^{17} (y^{\frac{17}{4}} a^{\frac{- 1 + 4}{3}})}$

Step 13.1.4

Add $- 1$ and $4$ .

$\frac{x}{b^{17} (y^{\frac{17}{4}} a^{\frac{3}{3}})}$

Step 13.1.5

Divide $3$ by $3$ .

$\frac{x}{b^{17} (y^{\frac{17}{4}} a^{1})}$

Step 13.2

Simplify $b^{17} (y^{\frac{17}{4}} a^{1})$ .

$\frac{x}{b^{17} (y^{\frac{17}{4}} a)}$

Step 14

Reorder factors in $\frac{x}{b^{17} y^{\frac{17}{4}} a}$ .

$\frac{x}{b^{17} a y^{\frac{17}{4}}}$