Precalculus Examples

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

Step 1

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

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Step 1.1

Apply the product rule to $\frac{4 y^{3} z^{\frac{2}{3}}}{x^{\frac{1}{2}}}$ .

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

Step 1.2

Apply the product rule to $4 y^{3} z^{\frac{2}{3}}$ .

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

Step 1.3

Apply the product rule to $4 y^{3}$ .

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

Step 2

Simplify the numerator.

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Step 2.1

Raise $4$ to the power of $2$ .

$\frac{16 {(y^{3})}^{2} {(z^{\frac{2}{3}})}^{2}}{{(x^{\frac{1}{2}})}^{2}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 2.2

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

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Step 2.2.1

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

$\frac{16 y^{3 \cdot 2} {(z^{\frac{2}{3}})}^{2}}{{(x^{\frac{1}{2}})}^{2}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 2.2.2

Multiply $3$ by $2$ .

$\frac{16 y^{6} {(z^{\frac{2}{3}})}^{2}}{{(x^{\frac{1}{2}})}^{2}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 2.3

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

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Step 2.3.1

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

$\frac{16 y^{6} z^{\frac{2}{3} \cdot 2}}{{(x^{\frac{1}{2}})}^{2}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 2.3.2

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

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Step 2.3.2.1

Combine $\frac{2}{3}$ and $2$ .

$\frac{16 y^{6} z^{\frac{2 \cdot 2}{3}}}{{(x^{\frac{1}{2}})}^{2}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 2.3.2.2

Multiply $2$ by $2$ .

$\frac{16 y^{6} z^{\frac{4}{3}}}{{(x^{\frac{1}{2}})}^{2}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 3

Simplify the denominator.

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Step 3.1

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

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Step 3.1.1

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

$\frac{16 y^{6} z^{\frac{4}{3}}}{x^{\frac{1}{2} \cdot 2}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Cancel the common factor.

$\frac{16 y^{6} z^{\frac{4}{3}}}{x^{\frac{1}{2} \cdot 2}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 3.1.2.2

Rewrite the expression.

$\frac{16 y^{6} z^{\frac{4}{3}}}{x^{1}} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 3.2

Simplify.

$\frac{16 y^{6} z^{\frac{4}{3}}}{x} {(\frac{x^{- 3} y^{6}}{8 z^{4}})}^{\frac{1}{3}}$

Step 4

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

$\frac{16 y^{6} z^{\frac{4}{3}}}{x} {(\frac{y^{6}}{8 z^{4} x^{3}})}^{\frac{1}{3}}$

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 $\frac{y^{6}}{8 z^{4} x^{3}}$ .

$\frac{16 y^{6} z^{\frac{4}{3}}}{x} \cdot \frac{{(y^{6})}^{\frac{1}{3}}}{{(8 z^{4} x^{3})}^{\frac{1}{3}}}$

Step 5.2

Apply the product rule to $8 z^{4} x^{3}$ .

$\frac{16 y^{6} z^{\frac{4}{3}}}{x} \cdot \frac{{(y^{6})}^{\frac{1}{3}}}{{(8 z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 5.3

Apply the product rule to $8 z^{4}$ .

$\frac{16 y^{6} z^{\frac{4}{3}}}{x} \cdot \frac{{(y^{6})}^{\frac{1}{3}}}{8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 6

Combine.

$\frac{16 y^{6} z^{\frac{4}{3}} {(y^{6})}^{\frac{1}{3}}}{x (8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}})}$

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

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

$\frac{16 y^{6} z^{\frac{4}{3}} y^{6 (\frac{1}{3})}}{x (8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}})}$

Step 7.1.2

Cancel the common factor of $3$ .

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Step 7.1.2.1

Factor $3$ out of $6$ .

$\frac{16 y^{6} z^{\frac{4}{3}} y^{3 (2) \frac{1}{3}}}{x (8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}})}$

Step 7.1.2.2

Cancel the common factor.

$\frac{16 y^{6} z^{\frac{4}{3}} y^{3 \cdot 2 \frac{1}{3}}}{x (8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}})}$

Step 7.1.2.3

Rewrite the expression.

$\frac{16 y^{6} z^{\frac{4}{3}} y^{2}}{x (8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}})}$

Step 7.2

Multiply $y^{6}$ by $y^{2}$ by adding the exponents.

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Step 7.2.1

Move $y^{2}$ .

$\frac{16 (y^{2} y^{6}) z^{\frac{4}{3}}}{x (8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}})}$

Step 7.2.2

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

$\frac{16 y^{2 + 6} z^{\frac{4}{3}}}{x (8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}})}$

Step 7.2.3

Add $2$ and $6$ .

$\frac{16 y^{8} z^{\frac{4}{3}}}{x (8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}})}$

Step 8

Simplify the denominator.

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Step 8.1

Remove unnecessary parentheses.

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 8^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 8.2

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

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot {(2^{3})}^{\frac{1}{3}} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 8.3

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

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2^{3 (\frac{1}{3})} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 8.4

Cancel the common factor of $3$ .

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Step 8.4.1

Cancel the common factor.

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2^{3 (\frac{1}{3})} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 8.4.2

Rewrite the expression.

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2^{1} {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 8.5

Evaluate the exponent.

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2 {(z^{4})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 8.6

Multiply the exponents in ${(z^{4})}^{\frac{1}{3}}$ .

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Step 8.6.1

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

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2 z^{4 (\frac{1}{3})} {(x^{3})}^{\frac{1}{3}}}$

Step 8.6.2

Combine $4$ and $\frac{1}{3}$ .

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2 z^{\frac{4}{3}} {(x^{3})}^{\frac{1}{3}}}$

Step 8.7

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

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Step 8.7.1

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

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2 z^{\frac{4}{3}} x^{3 (\frac{1}{3})}}$

Step 8.7.2

Cancel the common factor of $3$ .

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Step 8.7.2.1

Cancel the common factor.

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2 z^{\frac{4}{3}} x^{3 (\frac{1}{3})}}$

Step 8.7.2.2

Rewrite the expression.

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2 z^{\frac{4}{3}} x^{1}}$

Step 8.8

Simplify.

$\frac{16 y^{8} z^{\frac{4}{3}}}{x \cdot 2 z^{\frac{4}{3}} x}$

Step 8.9

Combine exponents.

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Step 8.9.1

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

$\frac{16 y^{8} z^{\frac{4}{3}}}{x^{1} x \cdot 2 z^{\frac{4}{3}}}$

Step 8.9.2

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

$\frac{16 y^{8} z^{\frac{4}{3}}}{x^{1} x^{1} \cdot 2 z^{\frac{4}{3}}}$

Step 8.9.3

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

$\frac{16 y^{8} z^{\frac{4}{3}}}{x^{1 + 1} \cdot 2 z^{\frac{4}{3}}}$

Step 8.9.4

Add $1$ and $1$ .

$\frac{16 y^{8} z^{\frac{4}{3}}}{x^{2} \cdot 2 z^{\frac{4}{3}}}$

Step 9

Factor $2$ out of $16 y^{8} z^{\frac{4}{3}}$ .

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

Step 10

Cancel the common factors.

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Step 10.1

Factor $2$ out of $x^{2} \cdot 2 z^{\frac{4}{3}}$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

$\frac{8 y^{8} z^{\frac{4}{3}}}{x^{2} z^{\frac{4}{3}}}$

Step 11

Cancel the common factor.

$\frac{8 y^{8} z^{\frac{4}{3}}}{x^{2} z^{\frac{4}{3}}}$

Step 12

Rewrite the expression.

$\frac{8 y^{8}}{x^{2}}$