Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

Factor $y$ out of $y x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.2.1

Multiply $x$ by $x^{2}$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

Add $1$ and $2$ .

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Move the leading negative in $- \frac{y^{2}}{x}$ into the numerator.

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

Step 3.3.2

Factor $x$ out of $y x$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $2$ .

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Move $y$ .

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

Step 4.1.2

Multiply $y$ by $y$ .

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

Step 4.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.2.1

Move $x^{2}$ .

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

Step 4.2.2

Multiply $x^{2}$ by $x$ .

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.3

Add $2$ and $1$ .

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

Step 4.3

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 4.3.1

Move $y^{2}$ .

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

Step 4.3.2

Multiply $y^{2}$ by $y$ .

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Step 4.3.2.1

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

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

Step 4.3.2.2

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

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

Step 4.3.3

Add $2$ and $1$ .

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

Step 4.4

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

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Step 4.4.1

Move $x$ .

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

Step 4.4.2

Multiply $x$ by $x$ .

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

Step 4.5

Multiply $y$ by $y^{3}$ by adding the exponents.

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Step 4.5.1

Move $y^{3}$ .

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

Step 4.5.2

Multiply $y^{3}$ by $y$ .

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Step 4.5.2.1

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

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

Step 4.5.2.2

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

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

Step 4.5.3

Add $3$ and $1$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

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

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

Step 4.6.5

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

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

Step 5

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

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Reorder the terms.

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

Step 6.2

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

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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