Algebra Examples

$\frac{{(- 8 x y)}^{2} - x^{2} y^{2}}{- 9 y^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(- 8 x y)}^{2} - {(x y)}^{2}}{- 9 y^{3}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 8 x y$ and $b = x y$ .

$\frac{(- 8 x y + x y) (- 8 x y - (x y))}{- 9 y^{3}}$

Step 1.3

Simplify.

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Step 1.3.1

Add $- 8 x y$ and $x y$ .

$\frac{- 7 x y (- 8 x y - (x y))}{- 9 y^{3}}$

Step 1.3.2

Subtract $x y$ from $- 8 x y$ .

$\frac{- 7 x y \cdot - 9 x y}{- 9 y^{3}}$

Step 1.3.3

Combine exponents.

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Step 1.3.3.1

Multiply $- 9$ by $- 7$ .

$\frac{63 x y x y}{- 9 y^{3}}$

Step 1.3.3.2

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

$\frac{63 (x^{1} x) y \cdot y}{- 9 y^{3}}$

Step 1.3.3.3

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

$\frac{63 (x^{1} x^{1}) y \cdot y}{- 9 y^{3}}$

Step 1.3.3.4

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

$\frac{63 x^{1 + 1} y \cdot y}{- 9 y^{3}}$

Step 1.3.3.5

Add $1$ and $1$ .

$\frac{63 x^{2} y \cdot y}{- 9 y^{3}}$

Step 1.3.3.6

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

$\frac{63 x^{2} (y^{1} y)}{- 9 y^{3}}$

Step 1.3.3.7

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

$\frac{63 x^{2} (y^{1} y^{1})}{- 9 y^{3}}$

Step 1.3.3.8

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

$\frac{63 x^{2} y^{1 + 1}}{- 9 y^{3}}$

Step 1.3.3.9

Add $1$ and $1$ .

$\frac{63 x^{2} y^{2}}{- 9 y^{3}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $63$ and $- 9$ .

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Step 2.1.1

Factor $9$ out of $63 x^{2} y^{2}$ .

$\frac{9 (7 x^{2} y^{2})}{- 9 y^{3}}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $9$ out of $- 9 y^{3}$ .

$\frac{9 (7 x^{2} y^{2})}{9 (- y^{3})}$

Step 2.1.2.2

Cancel the common factor.

$\frac{9 (7 x^{2} y^{2})}{9 (- y^{3})}$

Step 2.1.2.3

Rewrite the expression.

$\frac{7 x^{2} y^{2}}{- y^{3}}$

Step 2.2

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

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Step 2.2.1

Factor $y^{2}$ out of $7 x^{2} y^{2}$ .

$\frac{y^{2} (7 x^{2})}{- y^{3}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

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

$\frac{y^{2} (7 x^{2})}{y^{2} (- y)}$

Step 2.2.2.2

Cancel the common factor.

$\frac{y^{2} (7 x^{2})}{y^{2} (- y)}$

Step 2.2.2.3

Rewrite the expression.

$\frac{7 x^{2}}{- y}$

Step 2.3

Move the negative in front of the fraction.

$- \frac{7 x^{2}}{y}$