Algebra Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

Factor $y^{- 1} x$ out of $3 y^{2} x - x^{2} y^{- 1}$ .

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Step 2.1.1

Factor $y^{- 1} x$ out of $3 y^{2} x$ .

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

Step 2.1.2

Factor $y^{- 1} x$ out of $- x^{2} y^{- 1}$ .

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

Step 2.1.3

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

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

Step 2.2

Apply the product rule to $y^{- 1} x (3 y^{3} - x)$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Apply the product rule to $\frac{x}{y}$ .

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

Step 2.6

Combine exponents.

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.7

Reduce the expression by cancelling the common factors.

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Step 2.7.1

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

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Step 2.7.1.1

Cancel the common factor.

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

Step 2.7.1.2

Rewrite the expression.

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

Step 2.7.2

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

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

Step 3

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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