Precalculus Examples

$\frac{{(32 x^{5} y^{- \frac{3}{2}})}^{\frac{2}{5}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(32 x^{5} \frac{1}{y^{\frac{3}{2}}})}^{\frac{2}{5}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.2

Multiply $32 x^{5} \frac{1}{y^{\frac{3}{2}}}$ .

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Step 1.2.1

Combine $32$ and $\frac{1}{y^{\frac{3}{2}}}$ .

$\frac{{(x^{5} \frac{32}{y^{\frac{3}{2}}})}^{\frac{2}{5}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.2.2

Combine $x^{5}$ and $\frac{32}{y^{\frac{3}{2}}}$ .

$\frac{{(\frac{x^{5} \cdot 32}{y^{\frac{3}{2}}})}^{\frac{2}{5}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.3

Move $32$ to the left of $x^{5}$ .

$\frac{{(\frac{32 \cdot x^{5}}{y^{\frac{3}{2}}})}^{\frac{2}{5}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.4

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

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Step 1.4.1

Apply the product rule to $\frac{32 x^{5}}{y^{\frac{3}{2}}}$ .

$\frac{\frac{{(32 x^{5})}^{\frac{2}{5}}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.4.2

Apply the product rule to $32 x^{5}$ .

$\frac{\frac{32^{\frac{2}{5}} {(x^{5})}^{\frac{2}{5}}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

$\frac{\frac{{(2^{5})}^{\frac{2}{5}} {(x^{5})}^{\frac{2}{5}}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.5.2

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

$\frac{\frac{2^{5 (\frac{2}{5})} {(x^{5})}^{\frac{2}{5}}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.5.3

Cancel the common factor of $5$ .

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Step 1.5.3.1

Cancel the common factor.

$\frac{\frac{2^{5 (\frac{2}{5})} {(x^{5})}^{\frac{2}{5}}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.5.3.2

Rewrite the expression.

$\frac{\frac{2^{2} {(x^{5})}^{\frac{2}{5}}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.5.4

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

$\frac{\frac{4 {(x^{5})}^{\frac{2}{5}}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.5.5

Multiply the exponents in ${(x^{5})}^{\frac{2}{5}}$ .

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Step 1.5.5.1

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

$\frac{\frac{4 x^{5 (\frac{2}{5})}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.5.5.2

Cancel the common factor of $5$ .

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Step 1.5.5.2.1

Cancel the common factor.

$\frac{\frac{4 x^{5 (\frac{2}{5})}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.5.5.2.2

Rewrite the expression.

$\frac{\frac{4 x^{2}}{{(y^{\frac{3}{2}})}^{\frac{2}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 1.6

Multiply the exponents in ${(y^{\frac{3}{2}})}^{\frac{2}{5}}$ .

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Step 1.6.1

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

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

Step 1.6.2

Cancel the common factor of $2$ .

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Step 1.6.2.1

Cancel the common factor.

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

Step 1.6.2.2

Rewrite the expression.

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

Step 1.6.3

Combine $3$ and $\frac{1}{5}$ .

$\frac{\frac{4 x^{2}}{y^{\frac{3}{5}}}}{{(x^{\frac{5}{3}} y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $x^{\frac{5}{3}} y^{\frac{2}{3}}$ .

$\frac{\frac{4 x^{2}}{y^{\frac{3}{5}}}}{{(x^{\frac{5}{3}})}^{\frac{3}{5}} {(y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 2.2

Multiply the exponents in ${(x^{\frac{5}{3}})}^{\frac{3}{5}}$ .

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor of $5$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

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

Step 2.2.3

Cancel the common factor of $3$ .

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Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Rewrite the expression.

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

Step 2.3

Simplify.

$\frac{\frac{4 x^{2}}{y^{\frac{3}{5}}}}{x {(y^{\frac{2}{3}})}^{\frac{3}{5}}}$

Step 2.4

Multiply the exponents in ${(y^{\frac{2}{3}})}^{\frac{3}{5}}$ .

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Step 2.4.1

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

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

Step 2.4.2

Cancel the common factor of $3$ .

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Step 2.4.2.1

Cancel the common factor.

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

Step 2.4.2.2

Rewrite the expression.

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

Step 2.4.3

Combine $2$ and $\frac{1}{5}$ .

$\frac{\frac{4 x^{2}}{y^{\frac{3}{5}}}}{x y^{\frac{2}{5}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine.

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

Step 5

Multiply $y^{\frac{3}{5}}$ by $y^{\frac{2}{5}}$ by adding the exponents.

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Step 5.1

Move $y^{\frac{2}{5}}$ .

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

Step 5.2

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

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

Add $2$ and $3$ .

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

Step 5.5

Divide $5$ by $5$ .

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

Step 6

Simplify the expression.

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Step 6.1

Simplify $y^{1} x$ .

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

Step 6.2

Multiply $4 x^{2}$ by $1$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $x$ out of $y x$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$\frac{4 x}{y}$