Algebra Examples

$x^{- 3} y^{2} (y x^{4} + y^{- 1} x^{3} + y^{- 2} x^{2})$

Step 1

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

$\frac{1}{x^{3}} y^{2} (y x^{4} + y^{- 1} x^{3} + y^{- 2} x^{2})$

Step 2

Simplify terms.

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Step 2.1

Combine $\frac{1}{x^{3}}$ and $y^{2}$ .

$\frac{y^{2}}{x^{3}} (y x^{4} + y^{- 1} x^{3} + y^{- 2} x^{2})$

Step 2.2

Simplify each term.

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Step 2.2.1

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

$\frac{y^{2}}{x^{3}} (y x^{4} + \frac{1}{y} x^{3} + y^{- 2} x^{2})$

Step 2.2.2

Combine $\frac{1}{y}$ and $x^{3}$ .

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

Step 2.2.3

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

$\frac{y^{2}}{x^{3}} (y x^{4} + \frac{x^{3}}{y} + \frac{1}{y^{2}} x^{2})$

Step 2.2.4

Combine $\frac{1}{y^{2}}$ and $x^{2}$ .

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

Step 2.3

Multiply $\frac{y^{2}}{x^{3}}$ by $y x^{4} + \frac{x^{3}}{y} + \frac{x^{2}}{y^{2}}$ .

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

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Factor $x^{2}$ out of $x^{2} (y x^{2} + \frac{x}{y}) + x^{2} \frac{1}{y^{2}}$ .

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

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

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Step 3.4.1.1

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

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

Step 3.4.1.2

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

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

Step 3.4.1.3

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

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

Step 3.4.1.4

Factor $x$ out of $x (y x y) + x \cdot 1$ .

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

Step 3.4.2

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

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Step 3.4.2.1

Move $y$ .

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

Step 3.4.2.2

Multiply $y$ by $y$ .

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

Step 3.5

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

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

Multiply $y$ by $y$ .

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

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

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Step 3.8.2.1

Move $x$ .

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

Step 3.8.2.2

Multiply $x$ by $x$ .

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

Step 3.8.3

Multiply $x$ by $1$ .

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

Step 3.8.4

Apply the distributive property.

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

Step 3.8.5

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

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Step 3.8.5.1

Move $y$ .

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

Step 3.8.5.2

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

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Step 3.8.5.2.1

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

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

Step 3.8.5.2.2

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

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

Step 3.8.5.3

Add $1$ and $2$ .

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

Step 3.9

Combine exponents.

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Step 3.9.1

Combine $y^{2}$ and $\frac{x^{2} y^{3} + x y + 1}{y^{2}}$ .

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

Step 3.9.2

Combine $x^{2}$ and $\frac{y^{2} (x^{2} y^{3} + x y + 1)}{y^{2}}$ .

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

Step 3.10

Remove unnecessary parentheses.

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

Step 3.11

Reduce the expression by cancelling the common factors.

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Step 3.11.1

Reduce the expression $\frac{x^{2} y^{2} (x^{2} y^{3} + x y + 1)}{y^{2}}$ by cancelling the common factors.

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Step 3.11.1.1

Cancel the common factor.

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

Step 3.11.1.2

Rewrite the expression.

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

Step 3.11.2

Divide $x^{2} (x^{2} y^{3} + x y + 1)$ by $1$ .

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

Step 4

Cancel the common factors.

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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