Algebra Examples

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

Step 1

Move $x^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$(x^{- 2} - y^{- 2}) x^{2} y^{2}$

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

To write $\frac{1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y^{2}}{y^{2}}$ .

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

Step 5

To write $- \frac{1}{y^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Step 6

Write each expression with a common denominator of $x^{2} y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Reorder the factors of $y^{2} x^{2}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

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

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

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Cancel the common factor of $x^{2}$ .

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Step 9.1.1

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

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

Step 9.1.2

Cancel the common factor.

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

Step 9.1.3

Rewrite the expression.

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

Step 9.2

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

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

$(y + x) (y - x)$

Step 10

Expand $(y + x) (y - x)$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

$y (y - x) + x (y - x)$

Step 10.2

Apply the distributive property.

$y \cdot y + y (- x) + x (y - x)$

Step 10.3

Apply the distributive property.

$y \cdot y + y (- x) + x y + x (- x)$

Step 11

Simplify terms.

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Step 11.1

Combine the opposite terms in $y \cdot y + y (- x) + x y + x (- x)$ .

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Step 11.1.1

Reorder the factors in the terms $y (- x)$ and $x y$ .

$y \cdot y - x y + x y + x (- x)$

Step 11.1.2

Add $- x y$ and $x y$ .

$y \cdot y + 0 + x (- x)$

Step 11.1.3

Add $y \cdot y$ and $0$ .

$y \cdot y + x (- x)$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply $y$ by $y$ .

$y^{2} + x (- x)$

Step 11.2.2

Rewrite using the commutative property of multiplication.

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

Step 11.2.3

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

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Step 11.2.3.1

Move $x$ .

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

Step 11.2.3.2

Multiply $x$ by $x$ .

$y^{2} - x^{2}$