Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $x$ .

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

Cancel the common factor.

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

Step 1.1.4

Rewrite the expression.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Add $1$ and $2$ .

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

Step 1.5

Rewrite $- 1 (x y^{3})$ as $- (x y^{3})$ .

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

Step 2

To write $- x y^{3}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine $- x y^{3}$ and $\frac{y}{y}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

$\frac{x^{1} - x y^{3} y}{y}$

Step 4.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 - x y^{3} y}{y}$

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.2

Rewrite $1$ as $1^{2}$ .

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

Step 4.3

Rewrite $y^{3} y$ as ${(y^{2})}^{2}$ .

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

Step 4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = y^{2}$ .

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

Step 4.5

Simplify.

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Step 4.5.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = y$ .

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

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