Finite Math Examples

$\frac{{(x^{3} y^{2})}^{\frac{1}{6}}}{{(x^{- 7} y^{- 1})}^{- \frac{1}{3}}}$

Step 1

Move ${(x^{- 7} y^{- 1})}^{- \frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(x^{3} y^{2})}^{\frac{1}{6}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 2

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

${(x^{3})}^{\frac{1}{6}} {(y^{2})}^{\frac{1}{6}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 3

Multiply the exponents in ${(x^{3})}^{\frac{1}{6}}$ .

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Step 3.1

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

$x^{3 (\frac{1}{6})} {(y^{2})}^{\frac{1}{6}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Factor $3$ out of $6$ .

$x^{3 \frac{1}{3 (2)}} {(y^{2})}^{\frac{1}{6}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 3.2.2

Cancel the common factor.

$x^{3 \frac{1}{3 \cdot 2}} {(y^{2})}^{\frac{1}{6}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 3.2.3

Rewrite the expression.

$x^{\frac{1}{2}} {(y^{2})}^{\frac{1}{6}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 4

Multiply the exponents in ${(y^{2})}^{\frac{1}{6}}$ .

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Step 4.1

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

$x^{\frac{1}{2}} y^{2 (\frac{1}{6})} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $6$ .

$x^{\frac{1}{2}} y^{2 \frac{1}{2 (3)}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 4.2.2

Cancel the common factor.

$x^{\frac{1}{2}} y^{2 \frac{1}{2 \cdot 3}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 4.2.3

Rewrite the expression.

$x^{\frac{1}{2}} y^{\frac{1}{3}} {(x^{- 7} y^{- 1})}^{\frac{1}{3}}$

Step 5

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

$x^{\frac{1}{2}} y^{\frac{1}{3}} {(\frac{1}{x^{7}} y^{- 1})}^{\frac{1}{3}}$

Step 6

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

$x^{\frac{1}{2}} y^{\frac{1}{3}} {(\frac{1}{x^{7}} \cdot \frac{1}{y})}^{\frac{1}{3}}$

Step 7

Combine.

$x^{\frac{1}{2}} y^{\frac{1}{3}} {(\frac{1 \cdot 1}{x^{7} y})}^{\frac{1}{3}}$

Step 8

Multiply $1$ by $1$ .

$x^{\frac{1}{2}} y^{\frac{1}{3}} {(\frac{1}{x^{7} y})}^{\frac{1}{3}}$

Step 9

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

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Step 9.1

Apply the product rule to $\frac{1}{x^{7} y}$ .

$x^{\frac{1}{2}} y^{\frac{1}{3}} \frac{1^{\frac{1}{3}}}{{(x^{7} y)}^{\frac{1}{3}}}$

Step 9.2

Apply the product rule to $x^{7} y$ .

$x^{\frac{1}{2}} y^{\frac{1}{3}} \frac{1^{\frac{1}{3}}}{{(x^{7})}^{\frac{1}{3}} y^{\frac{1}{3}}}$

Step 10

Cancel the common factor of $y^{\frac{1}{3}}$ .

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Step 10.1

Factor $y^{\frac{1}{3}}$ out of $x^{\frac{1}{2}} y^{\frac{1}{3}}$ .

$y^{\frac{1}{3}} x^{\frac{1}{2}} \frac{1^{\frac{1}{3}}}{{(x^{7})}^{\frac{1}{3}} y^{\frac{1}{3}}}$

Step 10.2

Factor $y^{\frac{1}{3}}$ out of ${(x^{7})}^{\frac{1}{3}} y^{\frac{1}{3}}$ .

$y^{\frac{1}{3}} x^{\frac{1}{2}} \frac{1^{\frac{1}{3}}}{y^{\frac{1}{3}} {(x^{7})}^{\frac{1}{3}}}$

Step 10.3

Cancel the common factor.

$y^{\frac{1}{3}} x^{\frac{1}{2}} \frac{1^{\frac{1}{3}}}{y^{\frac{1}{3}} {(x^{7})}^{\frac{1}{3}}}$

Step 10.4

Rewrite the expression.

$x^{\frac{1}{2}} \frac{1^{\frac{1}{3}}}{{(x^{7})}^{\frac{1}{3}}}$

Step 11

Combine $x^{\frac{1}{2}}$ and $\frac{1^{\frac{1}{3}}}{{(x^{7})}^{\frac{1}{3}}}$ .

$\frac{x^{\frac{1}{2}} \cdot 1^{\frac{1}{3}}}{{(x^{7})}^{\frac{1}{3}}}$

Step 12

Simplify the expression.

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Step 12.1

One to any power is one.

$\frac{x^{\frac{1}{2}} \cdot 1}{{(x^{7})}^{\frac{1}{3}}}$

Step 12.2

Multiply the exponents in ${(x^{7})}^{\frac{1}{3}}$ .

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Step 12.2.1

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

$\frac{x^{\frac{1}{2}} \cdot 1}{x^{7 (\frac{1}{3})}}$

Step 12.2.2

Combine $7$ and $\frac{1}{3}$ .

$\frac{x^{\frac{1}{2}} \cdot 1}{x^{\frac{7}{3}}}$

Step 12.3

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

$\frac{x^{\frac{1}{2}}}{x^{\frac{7}{3}}}$

Step 12.4

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

$\frac{1}{x^{\frac{7}{3}} x^{- \frac{1}{2}}}$

Step 13

Multiply $x^{\frac{7}{3}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 13.1

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

$\frac{1}{x^{\frac{7}{3} - \frac{1}{2}}}$

Step 13.2

To write $\frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{x^{\frac{7}{3} \cdot \frac{2}{2} - \frac{1}{2}}}$

Step 13.3

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{x^{\frac{7}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}}}$

Step 13.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.4.1

Multiply $\frac{7}{3}$ by $\frac{2}{2}$ .

$\frac{1}{x^{\frac{7 \cdot 2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}}}$

Step 13.4.2

Multiply $3$ by $2$ .

$\frac{1}{x^{\frac{7 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3}}}$

Step 13.4.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\frac{1}{x^{\frac{7 \cdot 2}{6} - \frac{3}{2 \cdot 3}}}$

Step 13.4.4

Multiply $2$ by $3$ .

$\frac{1}{x^{\frac{7 \cdot 2}{6} - \frac{3}{6}}}$

Step 13.5

Combine the numerators over the common denominator.

$\frac{1}{x^{\frac{7 \cdot 2 - 3}{6}}}$

Step 13.6

Simplify the numerator.

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Step 13.6.1

Multiply $7$ by $2$ .

$\frac{1}{x^{\frac{14 - 3}{6}}}$

Step 13.6.2

Subtract $3$ from $14$ .

$\frac{1}{x^{\frac{11}{6}}}$