Calculus Examples

${(\frac{x^{2 n - 2} y^{2 n}}{x^{5 n + 1} y^{- n}})}^{\frac{1}{3}}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Cancel the common factor of $x^{2 n - 2}$ and $x^{5 n + 1}$ .

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Step 1.1.1

Factor $x^{5 n + 1}$ out of $x^{2 n - 2} y^{2 n}$ .

${(\frac{x^{5 n + 1} (x^{2 n - 2 - (5 n + 1)} y^{2 n})}{x^{5 n + 1} y^{- n}})}^{\frac{1}{3}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $x^{5 n + 1}$ out of $x^{5 n + 1} y^{- n}$ .

${(\frac{x^{5 n + 1} (x^{2 n - 2 - (5 n + 1)} y^{2 n})}{x^{5 n + 1} (y^{- n})})}^{\frac{1}{3}}$

Step 1.1.2.2

Cancel the common factor.

${(\frac{x^{5 n + 1} (x^{2 n - 2 - (5 n + 1)} y^{2 n})}{x^{5 n + 1} y^{- n}})}^{\frac{1}{3}}$

Step 1.1.2.3

Rewrite the expression.

${(\frac{x^{2 n - 2 - (5 n + 1)} y^{2 n}}{y^{- n}})}^{\frac{1}{3}}$

Step 1.2

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

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Step 1.2.1

Factor $y^{- n}$ out of $x^{2 n - 2 - (5 n + 1)} y^{2 n}$ .

${(\frac{y^{- n} (x^{2 n - 2 - (5 n + 1)} y^{3 n})}{y^{- n}})}^{\frac{1}{3}}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Multiply by $1$ .

${(\frac{y^{- n} (x^{2 n - 2 - (5 n + 1)} y^{3 n})}{y^{- n} \cdot 1})}^{\frac{1}{3}}$

Step 1.2.2.2

Cancel the common factor.

${(\frac{y^{- n} (x^{2 n - 2 - (5 n + 1)} y^{3 n})}{y^{- n} \cdot 1})}^{\frac{1}{3}}$

Step 1.2.2.3

Rewrite the expression.

${(\frac{x^{2 n - 2 - (5 n + 1)} y^{3 n}}{1})}^{\frac{1}{3}}$

Step 1.2.2.4

Divide $x^{2 n - 2 - (5 n + 1)} y^{3 n}$ by $1$ .

${(x^{2 n - 2 - (5 n + 1)} y^{3 n})}^{\frac{1}{3}}$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $5$ by $- 1$ .

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

Step 2.3

Multiply $- 1$ by $1$ .

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

Step 3

Simplify by adding terms.

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Step 3.1

Subtract $5 n$ from $2 n$ .

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

Step 3.2

Simplify the expression.

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Step 3.2.1

Subtract $1$ from $- 2$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

Apply the distributive property.

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

Step 3.2.3.3

Cancel the common factor of $3$ .

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Step 3.2.3.3.1

Factor $3$ out of $- 3 n$ .

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

Step 3.2.3.3.2

Cancel the common factor.

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

Step 3.2.3.3.3

Rewrite the expression.

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

Step 3.2.3.4

Cancel the common factor of $3$ .

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Step 3.2.3.4.1

Factor $3$ out of $- 3$ .

$x^{- n + 3 (- 1) \frac{1}{3}} {(y^{3 n})}^{\frac{1}{3}}$

Step 3.2.3.4.2

Cancel the common factor.

$x^{- n + 3 \cdot - 1 \frac{1}{3}} {(y^{3 n})}^{\frac{1}{3}}$

Step 3.2.3.4.3

Rewrite the expression.

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

Step 3.2.4

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

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Step 3.2.4.1

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

$x^{- n - 1} y^{3 n \frac{1}{3}}$

Step 3.2.4.2

Cancel the common factor of $3$ .

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Step 3.2.4.2.1

Factor $3$ out of $3 n$ .

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

Step 3.2.4.2.2

Cancel the common factor.

$x^{- n - 1} y^{3 n \frac{1}{3}}$

Step 3.2.4.2.3

Rewrite the expression.

$x^{- n - 1} y^{n}$