Finite Math Examples

${(16 x^{\frac{1}{2}} y^{- \frac{1}{3}})}^{\frac{1}{2}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 1

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

${(16 x^{\frac{1}{2}} \frac{1}{y^{\frac{1}{3}}})}^{\frac{1}{2}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 2

Multiply $16 x^{\frac{1}{2}} \frac{1}{y^{\frac{1}{3}}}$ .

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Step 2.1

Combine $16$ and $\frac{1}{y^{\frac{1}{3}}}$ .

${(x^{\frac{1}{2}} \frac{16}{y^{\frac{1}{3}}})}^{\frac{1}{2}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 2.2

Combine $x^{\frac{1}{2}}$ and $\frac{16}{y^{\frac{1}{3}}}$ .

${(\frac{x^{\frac{1}{2}} \cdot 16}{y^{\frac{1}{3}}})}^{\frac{1}{2}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 3

Move $16$ to the left of $x^{\frac{1}{2}}$ .

${(\frac{16 \cdot x^{\frac{1}{2}}}{y^{\frac{1}{3}}})}^{\frac{1}{2}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{16 x^{\frac{1}{2}}}{y^{\frac{1}{3}}}$ .

$\frac{{(16 x^{\frac{1}{2}})}^{\frac{1}{2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 4.2

Apply the product rule to $16 x^{\frac{1}{2}}$ .

$\frac{16^{\frac{1}{2}} {(x^{\frac{1}{2}})}^{\frac{1}{2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $16$ as $4^{2}$ .

$\frac{{(4^{2})}^{\frac{1}{2}} {(x^{\frac{1}{2}})}^{\frac{1}{2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 5.2

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

$\frac{4^{2 (\frac{1}{2})} {(x^{\frac{1}{2}})}^{\frac{1}{2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Cancel the common factor.

$\frac{4^{2 (\frac{1}{2})} {(x^{\frac{1}{2}})}^{\frac{1}{2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 5.3.2

Rewrite the expression.

$\frac{4^{1} {(x^{\frac{1}{2}})}^{\frac{1}{2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 5.4

Evaluate the exponent.

$\frac{4 {(x^{\frac{1}{2}})}^{\frac{1}{2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 5.5

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

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Step 5.5.1

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

$\frac{4 x^{\frac{1}{2} \cdot \frac{1}{2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 5.5.2

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 5.5.2.1

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

$\frac{4 x^{\frac{1}{2 \cdot 2}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 5.5.2.2

Multiply $2$ by $2$ .

$\frac{4 x^{\frac{1}{4}}}{{(y^{\frac{1}{3}})}^{\frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 6

Multiply the exponents in ${(y^{\frac{1}{3}})}^{\frac{1}{2}}$ .

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Step 6.1

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

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{3} \cdot \frac{1}{2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 6.2

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

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Step 6.2.1

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

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{3 \cdot 2}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 6.2.2

Multiply $3$ by $2$ .

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} {(- 8 x^{- \frac{1}{6}} y)}^{- \frac{2}{3}}$

Step 7

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

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} {(- 8 \frac{1}{x^{\frac{1}{6}}} y)}^{- \frac{2}{3}}$

Step 8

Combine $- 8$ and $\frac{1}{x^{\frac{1}{6}}}$ .

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} {(\frac{- 8}{x^{\frac{1}{6}}} y)}^{- \frac{2}{3}}$

Step 9

Move the negative in front of the fraction.

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} {(- \frac{8}{x^{\frac{1}{6}}} y)}^{- \frac{2}{3}}$

Step 10

Combine $y$ and $\frac{8}{x^{\frac{1}{6}}}$ .

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} {(- \frac{y \cdot 8}{x^{\frac{1}{6}}})}^{- \frac{2}{3}}$

Step 11

Move $8$ to the left of $y$ .

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} {(- \frac{8 \cdot y}{x^{\frac{1}{6}}})}^{- \frac{2}{3}}$

Step 12

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} {(- \frac{x^{\frac{1}{6}}}{8 y})}^{\frac{2}{3}}$

Step 13

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

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Step 13.1

Apply the product rule to $- \frac{x^{\frac{1}{6}}}{8 y}$ .

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} ({(- 1)}^{\frac{2}{3}} {(\frac{x^{\frac{1}{6}}}{8 y})}^{\frac{2}{3}})$

Step 13.2

Apply the product rule to $\frac{x^{\frac{1}{6}}}{8 y}$ .

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} ({(- 1)}^{\frac{2}{3}} \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{{(8 y)}^{\frac{2}{3}}})$

Step 13.3

Apply the product rule to $8 y$ .

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} ({(- 1)}^{\frac{2}{3}} \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{8^{\frac{2}{3}} y^{\frac{2}{3}}})$

Step 14

Simplify the expression.

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Step 14.1

Rewrite using the commutative property of multiplication.

${(- 1)}^{\frac{2}{3}} \frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{8^{\frac{2}{3}} y^{\frac{2}{3}}}$

Step 14.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

${({(- 1)}^{3})}^{\frac{2}{3}} \frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{8^{\frac{2}{3}} y^{\frac{2}{3}}}$

Step 14.3

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

${(- 1)}^{3 (\frac{2}{3})} \frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{8^{\frac{2}{3}} y^{\frac{2}{3}}}$

Step 15

Cancel the common factor of $3$ .

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Step 15.1

Cancel the common factor.

${(- 1)}^{3 (\frac{2}{3})} \frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{8^{\frac{2}{3}} y^{\frac{2}{3}}}$

Step 15.2

Rewrite the expression.

${(- 1)}^{2} \frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{8^{\frac{2}{3}} y^{\frac{2}{3}}}$

Step 16

Simplify the expression.

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Step 16.1

Raise $- 1$ to the power of $2$ .

$1 \frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{8^{\frac{2}{3}} y^{\frac{2}{3}}}$

Step 16.2

Multiply $\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}}$ by $1$ .

$\frac{4 x^{\frac{1}{4}}}{y^{\frac{1}{6}}} \cdot \frac{{(x^{\frac{1}{6}})}^{\frac{2}{3}}}{8^{\frac{2}{3}} y^{\frac{2}{3}}}$

Step 17

Combine.

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{1}{6}} (8^{\frac{2}{3}} y^{\frac{2}{3}})}$

Step 18

Multiply $y^{\frac{1}{6}}$ by $y^{\frac{2}{3}}$ by adding the exponents.

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Step 18.1

Move $y^{\frac{2}{3}}$ .

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{2}{3}} y^{\frac{1}{6}} \cdot 8^{\frac{2}{3}}}$

Step 18.2

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

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{2}{3} + \frac{1}{6}} \cdot 8^{\frac{2}{3}}}$

Step 18.3

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

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{6}} \cdot 8^{\frac{2}{3}}}$

Step 18.4

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

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Step 18.4.1

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

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{2 \cdot 2}{3 \cdot 2} + \frac{1}{6}} \cdot 8^{\frac{2}{3}}}$

Step 18.4.2

Multiply $3$ by $2$ .

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{2 \cdot 2}{6} + \frac{1}{6}} \cdot 8^{\frac{2}{3}}}$

Step 18.5

Combine the numerators over the common denominator.

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{2 \cdot 2 + 1}{6}} \cdot 8^{\frac{2}{3}}}$

Step 18.6

Simplify the numerator.

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Step 18.6.1

Multiply $2$ by $2$ .

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{4 + 1}{6}} \cdot 8^{\frac{2}{3}}}$

Step 18.6.2

Add $4$ and $1$ .

$\frac{4 x^{\frac{1}{4}} {(x^{\frac{1}{6}})}^{\frac{2}{3}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19

Simplify the numerator.

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Step 19.1

Multiply the exponents in ${(x^{\frac{1}{6}})}^{\frac{2}{3}}$ .

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Step 19.1.1

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

$\frac{4 x^{\frac{1}{4}} x^{\frac{1}{6} \cdot \frac{2}{3}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.1.2

Cancel the common factor of $2$ .

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Step 19.1.2.1

Factor $2$ out of $6$ .

$\frac{4 x^{\frac{1}{4}} x^{\frac{1}{2 (3)} \cdot \frac{2}{3}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.1.2.2

Cancel the common factor.

$\frac{4 x^{\frac{1}{4}} x^{\frac{1}{2 \cdot 3} \cdot \frac{2}{3}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.1.2.3

Rewrite the expression.

$\frac{4 x^{\frac{1}{4}} x^{\frac{1}{3} \cdot \frac{1}{3}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.1.3

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

$\frac{4 x^{\frac{1}{4}} x^{\frac{1}{3 \cdot 3}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.1.4

Multiply $3$ by $3$ .

$\frac{4 x^{\frac{1}{4}} x^{\frac{1}{9}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2

Multiply $x^{\frac{1}{4}}$ by $x^{\frac{1}{9}}$ by adding the exponents.

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Step 19.2.1

Move $x^{\frac{1}{9}}$ .

$\frac{4 (x^{\frac{1}{9}} x^{\frac{1}{4}})}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.2

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

$\frac{4 x^{\frac{1}{9} + \frac{1}{4}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.3

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4 x^{\frac{1}{9} \cdot \frac{4}{4} + \frac{1}{4}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.4

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{4 x^{\frac{1}{9} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{9}{9}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.5

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 19.2.5.1

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$\frac{4 x^{\frac{4}{9 \cdot 4} + \frac{1}{4} \cdot \frac{9}{9}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.5.2

Multiply $9$ by $4$ .

$\frac{4 x^{\frac{4}{36} + \frac{1}{4} \cdot \frac{9}{9}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.5.3

Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .

$\frac{4 x^{\frac{4}{36} + \frac{9}{4 \cdot 9}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.5.4

Multiply $4$ by $9$ .

$\frac{4 x^{\frac{4}{36} + \frac{9}{36}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.6

Combine the numerators over the common denominator.

$\frac{4 x^{\frac{4 + 9}{36}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 19.2.7

Add $4$ and $9$ .

$\frac{4 x^{\frac{13}{36}}}{y^{\frac{5}{6}} \cdot 8^{\frac{2}{3}}}$

Step 20

Simplify the denominator.

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Step 20.1

Rewrite $8$ as $2^{3}$ .

$\frac{4 x^{\frac{13}{36}}}{y^{\frac{5}{6}} \cdot {(2^{3})}^{\frac{2}{3}}}$

Step 20.2

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

$\frac{4 x^{\frac{13}{36}}}{y^{\frac{5}{6}} \cdot 2^{3 (\frac{2}{3})}}$

Step 20.3

Cancel the common factor of $3$ .

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Step 20.3.1

Cancel the common factor.

$\frac{4 x^{\frac{13}{36}}}{y^{\frac{5}{6}} \cdot 2^{3 (\frac{2}{3})}}$

Step 20.3.2

Rewrite the expression.

$\frac{4 x^{\frac{13}{36}}}{y^{\frac{5}{6}} \cdot 2^{2}}$

Step 20.4

Raise $2$ to the power of $2$ .

$\frac{4 x^{\frac{13}{36}}}{y^{\frac{5}{6}} \cdot 4}$

Step 21

Reduce the expression by cancelling the common factors.

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Step 21.1

Cancel the common factor.

$\frac{4 x^{\frac{13}{36}}}{y^{\frac{5}{6}} \cdot 4}$

Step 21.2

Rewrite the expression.

$\frac{x^{\frac{13}{36}}}{y^{\frac{5}{6}}}$