Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Simplify the denominator.

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Step 3.1

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 3.2

To write $- \frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3.3

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Reorder the factors of $y x$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 7

Cancel the common factor of $x y$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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

Step 9

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

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Step 9.1

Factor $- 1$ out of $x$ .

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

Step 9.2

Factor $- 1$ out of $- y$ .

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

Step 9.3

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

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

Step 9.4

Reorder terms.

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

Step 9.5

Cancel the common factor.

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

Step 9.6

Divide $- 1$ by $1$ .

$- 1$