Precalculus Examples

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

Step 1

Move ${(x y^{4})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $4$ .

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

Step 2.3

Multiply the exponents in ${(y^{3})}^{4}$ .

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Step 2.3.1

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

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

Step 2.3.2

Multiply $3$ by $4$ .

$\frac{x^{8} y^{12}}{x^{2} y {(x y^{4})}^{3}}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{x^{8} y^{12}}{x^{2} y x^{3} {(y^{4})}^{3}}$

Step 3.2

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 3.2.1

Move $x^{3}$ .

$\frac{x^{8} y^{12}}{x^{3} x^{2} y {(y^{4})}^{3}}$

Step 3.2.2

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

$\frac{x^{8} y^{12}}{x^{3 + 2} y {(y^{4})}^{3}}$

Step 3.2.3

Add $3$ and $2$ .

$\frac{x^{8} y^{12}}{x^{5} y {(y^{4})}^{3}}$

Step 3.3

Multiply the exponents in ${(y^{4})}^{3}$ .

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Step 3.3.1

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

$\frac{x^{8} y^{12}}{x^{5} y \cdot y^{4 \cdot 3}}$

Step 3.3.2

Multiply $4$ by $3$ .

$\frac{x^{8} y^{12}}{x^{5} y \cdot y^{12}}$

Step 3.4

Multiply $y$ by $y^{12}$ by adding the exponents.

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Step 3.4.1

Move $y^{12}$ .

$\frac{x^{8} y^{12}}{x^{5} (y^{12} y)}$

Step 3.4.2

Multiply $y^{12}$ by $y$ .

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Step 3.4.2.1

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

$\frac{x^{8} y^{12}}{x^{5} (y^{12} y^{1})}$

Step 3.4.2.2

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

$\frac{x^{8} y^{12}}{x^{5} y^{12 + 1}}$

Step 3.4.3

Add $12$ and $1$ .

$\frac{x^{8} y^{12}}{x^{5} y^{13}}$

Step 4

Cancel the common factor of $x^{8}$ and $x^{5}$ .

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Step 4.1

Factor $x^{5}$ out of $x^{8} y^{12}$ .

$\frac{x^{5} (x^{3} y^{12})}{x^{5} y^{13}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $x^{5}$ out of $x^{5} y^{13}$ .

$\frac{x^{5} (x^{3} y^{12})}{x^{5} (y^{13})}$

Step 4.2.2

Cancel the common factor.

$\frac{x^{5} (x^{3} y^{12})}{x^{5} y^{13}}$

Step 4.2.3

Rewrite the expression.

$\frac{x^{3} y^{12}}{y^{13}}$

Step 5

Cancel the common factor of $y^{12}$ and $y^{13}$ .

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Step 5.1

Factor $y^{12}$ out of $x^{3} y^{12}$ .

$\frac{y^{12} x^{3}}{y^{13}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $y^{12}$ out of $y^{13}$ .

$\frac{y^{12} x^{3}}{y^{12} y}$

Step 5.2.2

Cancel the common factor.

$\frac{y^{12} x^{3}}{y^{12} y}$

Step 5.2.3

Rewrite the expression.

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