Finite Math Examples

$\frac{\frac{\frac{1}{x} - \frac{1}{y}}{x - y}}{x y}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{1}{x} - \frac{1}{y}}{x - y} \cdot \frac{1}{x y}$

Step 2

Multiply the numerator and denominator of the fraction by $x y$ .

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Step 2.1

Multiply $\frac{\frac{1}{x} - \frac{1}{y}}{x - y}$ by $\frac{x y}{x y}$ .

$\frac{x y}{x y} \cdot \frac{\frac{1}{x} - \frac{1}{y}}{x - y} \frac{1}{x y}$

Step 2.2

Combine.

$\frac{x y (\frac{1}{x} - \frac{1}{y})}{x y (x - y)} \cdot \frac{1}{x y}$

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Factor $x$ out of $x y$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $y$ .

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Step 4.2.1

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Step 4.2.2

Factor $y$ out of $x y$ .

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Move $- 1$ to the left of $x$ .

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

Step 5.2

Rewrite $- 1 x$ as $- x$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1.1

Move $x$ .

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

Step 6.1.2

Multiply $x$ by $x$ .

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

Step 6.2

Rewrite using the commutative property of multiplication.

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

Step 6.3

Multiply $y$ by $y$ by adding the exponents.

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Step 6.3.1

Move $y$ .

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

Step 6.3.2

Multiply $y$ by $y$ .

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

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

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

Step 6.4.3

Factor $x y$ out of $x y (x) + x y (- y)$ .

$\frac{y - x}{x y (x - y)} \cdot \frac{1}{x y}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $y - x$ and $x - y$ .

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Step 7.1.1

Factor $- 1$ out of $y$ .

$\frac{- 1 (- y) - x}{x y (x - y)} \cdot \frac{1}{x y}$

Step 7.1.2

Factor $- 1$ out of $- x$ .

$\frac{- 1 (- y) - (x)}{x y (x - y)} \cdot \frac{1}{x y}$

Step 7.1.3

Factor $- 1$ out of $- 1 (- y) - (x)$ .

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

Step 7.1.4

Reorder terms.

$\frac{- 1 (x - y)}{x y (x - y)} \cdot \frac{1}{x y}$

Step 7.1.5

Cancel the common factor.

$\frac{- 1 (x - y)}{x y (x - y)} \cdot \frac{1}{x y}$

Step 7.1.6

Rewrite the expression.

$\frac{- 1}{x y} \cdot \frac{1}{x y}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{1}{x y} \cdot \frac{1}{x y}$

Step 8

Multiply $- \frac{1}{x y} \cdot \frac{1}{x y}$ .

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Step 8.1

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

$- \frac{1}{x y (x y)}$

Step 8.2

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

$- \frac{1}{x^{1} x y \cdot y}$

Step 8.3

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

$- \frac{1}{x^{1} x^{1} y \cdot y}$

Step 8.4

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

$- \frac{1}{x^{1 + 1} y \cdot y}$

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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

Step 8.7

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

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

Step 8.8

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

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

Step 8.9

Add $1$ and $1$ .

$- \frac{1}{x^{2} y^{2}}$