Finite Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

Factor $x^{- 3} y^{- 2}$ out of $x^{- 3} y^{- 2} (x^{4}) + x^{- 3} y^{- 2} (y^{3})$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

Simplify the denominator.

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Step 2.1

Factor $x^{- 4} y^{- 1}$ out of $x^{2} y^{- 1} - x^{- 4} y^{3}$ .

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Step 2.1.1

Factor $x^{- 4} y^{- 1}$ out of $x^{2} y^{- 1}$ .

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

Step 2.1.2

Factor $x^{- 4} y^{- 1}$ out of $- x^{- 4} y^{3}$ .

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

Step 2.1.3

Factor $x^{- 4} y^{- 1}$ out of $x^{- 4} y^{- 1} (x^{6}) + x^{- 4} y^{- 1} (- 1 y^{4})$ .

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

Step 2.2

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

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

Step 2.3

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

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

Step 2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{3}$ and $b = y^{2}$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $\frac{1}{x^{4}}$ by $\frac{1}{y}$ .

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

Step 3

Simplify terms.

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Step 3.1

Factor $\frac{1}{x^{3} y^{2}}$ out of $\frac{\frac{1}{x^{3} y^{2}} (x^{4} + y^{3})}{\frac{1}{x^{4} y} (x^{3} + y^{2}) (x^{3} - y^{2})}$ .

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

Step 3.2

Factor $\frac{1}{x^{4} y}$ out of $\frac{x^{4} + y^{3}}{\frac{1}{x^{4} y} (x^{3} + y^{2}) (x^{3} - y^{2})}$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Factor $x^{3} y$ out of $x^{4} y \frac{x^{4} + y^{3}}{(x^{3} + y^{2}) (x^{3} - y^{2})}$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Combine $x$ and $\frac{x^{4} + y^{3}}{(x^{3} + y^{2}) (x^{3} - y^{2})}$ .

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

Step 3.5

Multiply $\frac{1}{y}$ by $\frac{x (x^{4} + y^{3})}{(x^{3} + y^{2}) (x^{3} - y^{2})}$ .

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