Precalculus Examples

$\frac{{(32 x^{- 10} y^{15})}^{\frac{1}{5}}}{{(64 x^{6} y^{- 12})}^{- \frac{1}{6}}}$

Step 1

Move ${(64 x^{6} y^{- 12})}^{- \frac{1}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(32 x^{- 10} y^{15})}^{\frac{1}{5}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 2

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

${(32 \frac{1}{x^{10}} y^{15})}^{\frac{1}{5}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 3

Combine $32$ and $\frac{1}{x^{10}}$ .

${(\frac{32}{x^{10}} y^{15})}^{\frac{1}{5}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 4

Combine $\frac{32}{x^{10}}$ and $y^{15}$ .

${(\frac{32 y^{15}}{x^{10}})}^{\frac{1}{5}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 5

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

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Step 5.1

Apply the product rule to $\frac{32 y^{15}}{x^{10}}$ .

$\frac{{(32 y^{15})}^{\frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 5.2

Apply the product rule to $32 y^{15}$ .

$\frac{32^{\frac{1}{5}} {(y^{15})}^{\frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $32$ as $2^{5}$ .

$\frac{{(2^{5})}^{\frac{1}{5}} {(y^{15})}^{\frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6.2

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

$\frac{2^{5 (\frac{1}{5})} {(y^{15})}^{\frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6.3

Cancel the common factor of $5$ .

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Step 6.3.1

Cancel the common factor.

$\frac{2^{5 (\frac{1}{5})} {(y^{15})}^{\frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6.3.2

Rewrite the expression.

$\frac{2^{1} {(y^{15})}^{\frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6.4

Evaluate the exponent.

$\frac{2 {(y^{15})}^{\frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6.5

Multiply the exponents in ${(y^{15})}^{\frac{1}{5}}$ .

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Step 6.5.1

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

$\frac{2 y^{15 (\frac{1}{5})}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6.5.2

Cancel the common factor of $5$ .

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Step 6.5.2.1

Factor $5$ out of $15$ .

$\frac{2 y^{5 (3) \frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6.5.2.2

Cancel the common factor.

$\frac{2 y^{5 \cdot 3 \frac{1}{5}}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 6.5.2.3

Rewrite the expression.

$\frac{2 y^{3}}{{(x^{10})}^{\frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 7

Multiply the exponents in ${(x^{10})}^{\frac{1}{5}}$ .

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Step 7.1

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

$\frac{2 y^{3}}{x^{10 (\frac{1}{5})}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 7.2

Cancel the common factor of $5$ .

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Step 7.2.1

Factor $5$ out of $10$ .

$\frac{2 y^{3}}{x^{5 (2) \frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 7.2.2

Cancel the common factor.

$\frac{2 y^{3}}{x^{5 \cdot 2 \frac{1}{5}}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 7.2.3

Rewrite the expression.

$\frac{2 y^{3}}{x^{2}} {(64 x^{6} y^{- 12})}^{\frac{1}{6}}$

Step 8

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

$\frac{2 y^{3}}{x^{2}} {(64 x^{6} \frac{1}{y^{12}})}^{\frac{1}{6}}$

Step 9

Multiply $64 x^{6} \frac{1}{y^{12}}$ .

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Step 9.1

Combine $64$ and $\frac{1}{y^{12}}$ .

$\frac{2 y^{3}}{x^{2}} {(x^{6} \frac{64}{y^{12}})}^{\frac{1}{6}}$

Step 9.2

Combine $x^{6}$ and $\frac{64}{y^{12}}$ .

$\frac{2 y^{3}}{x^{2}} {(\frac{x^{6} \cdot 64}{y^{12}})}^{\frac{1}{6}}$

Step 10

Move $64$ to the left of $x^{6}$ .

$\frac{2 y^{3}}{x^{2}} {(\frac{64 \cdot x^{6}}{y^{12}})}^{\frac{1}{6}}$

Step 11

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

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Step 11.1

Apply the product rule to $\frac{64 x^{6}}{y^{12}}$ .

$\frac{2 y^{3}}{x^{2}} \cdot \frac{{(64 x^{6})}^{\frac{1}{6}}}{{(y^{12})}^{\frac{1}{6}}}$

Step 11.2

Apply the product rule to $64 x^{6}$ .

$\frac{2 y^{3}}{x^{2}} \cdot \frac{64^{\frac{1}{6}} {(x^{6})}^{\frac{1}{6}}}{{(y^{12})}^{\frac{1}{6}}}$

Step 12

Combine.

$\frac{2 y^{3} (64^{\frac{1}{6}} {(x^{6})}^{\frac{1}{6}})}{x^{2} {(y^{12})}^{\frac{1}{6}}}$

Step 13

Simplify the numerator.

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Step 13.1

Rewrite $64$ as $2^{6}$ .

$\frac{{(2^{6})}^{\frac{1}{6}} \cdot 2 y^{3} {(x^{6})}^{\frac{1}{6}}}{x^{2} {(y^{12})}^{\frac{1}{6}}}$

Step 13.2

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

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Step 13.2.1

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

$\frac{2^{6 (\frac{1}{6})} \cdot 2 y^{3} {(x^{6})}^{\frac{1}{6}}}{x^{2} {(y^{12})}^{\frac{1}{6}}}$

Step 13.2.2

Cancel the common factor of $6$ .

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Step 13.2.2.1

Cancel the common factor.

$\frac{2^{6 (\frac{1}{6})} \cdot 2 y^{3} {(x^{6})}^{\frac{1}{6}}}{x^{2} {(y^{12})}^{\frac{1}{6}}}$

Step 13.2.2.2

Rewrite the expression.

$\frac{2^{1} \cdot 2 y^{3} {(x^{6})}^{\frac{1}{6}}}{x^{2} {(y^{12})}^{\frac{1}{6}}}$

Step 13.3

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

$\frac{2^{1 + 1} y^{3} {(x^{6})}^{\frac{1}{6}}}{x^{2} {(y^{12})}^{\frac{1}{6}}}$

Step 13.4

Add $1$ and $1$ .

$\frac{2^{2} y^{3} {(x^{6})}^{\frac{1}{6}}}{x^{2} {(y^{12})}^{\frac{1}{6}}}$

Step 14

Multiply the exponents in ${(y^{12})}^{\frac{1}{6}}$ .

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Step 14.1

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

$\frac{2^{2} y^{3} {(x^{6})}^{\frac{1}{6}}}{x^{2} y^{12 (\frac{1}{6})}}$

Step 14.2

Cancel the common factor of $6$ .

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Step 14.2.1

Factor $6$ out of $12$ .

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

Step 14.2.2

Cancel the common factor.

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

Step 14.2.3

Rewrite the expression.

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

Step 15

Simplify the numerator.

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Step 15.1

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

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

Step 15.2

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

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Step 15.2.1

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

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

Step 15.2.2

Cancel the common factor of $6$ .

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Step 15.2.2.1

Cancel the common factor.

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

Step 15.2.2.2

Rewrite the expression.

$\frac{4 y^{3} x^{1}}{x^{2} y^{2}}$

Step 15.3

Simplify.

$\frac{4 y^{3} x}{x^{2} y^{2}}$

Step 16

Reduce the expression by cancelling the common factors.

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Step 16.1

Cancel the common factor of $y^{3}$ and $y^{2}$ .

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Step 16.1.1

Factor $y^{2}$ out of $4 y^{3} x$ .

$\frac{y^{2} (4 y x)}{x^{2} y^{2}}$

Step 16.1.2

Cancel the common factors.

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Step 16.1.2.1

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

$\frac{y^{2} (4 y x)}{y^{2} x^{2}}$

Step 16.1.2.2

Cancel the common factor.

$\frac{y^{2} (4 y x)}{y^{2} x^{2}}$

Step 16.1.2.3

Rewrite the expression.

$\frac{4 y x}{x^{2}}$

Step 16.2

Cancel the common factor of $x$ and $x^{2}$ .

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Step 16.2.1

Factor $x$ out of $4 y x$ .

$\frac{x (4 y)}{x^{2}}$

Step 16.2.2

Cancel the common factors.

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Step 16.2.2.1

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

$\frac{x (4 y)}{x \cdot x}$

Step 16.2.2.2

Cancel the common factor.

$\frac{x (4 y)}{x \cdot x}$

Step 16.2.2.3

Rewrite the expression.

$\frac{4 y}{x}$