Pre-Algebra Examples

$\frac{9 x^{5}}{a^{2} - b^{2}} \div \frac{27 x^{2}}{3 b - 3 a}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{9 x^{5}}{a^{2} - b^{2}} \cdot \frac{3 b - 3 a}{27 x^{2}}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $9 x^{2}$ .

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Step 2.1.1

Factor $9 x^{2}$ out of $9 x^{5}$ .

$\frac{9 x^{2} (x^{3})}{a^{2} - b^{2}} \cdot \frac{3 b - 3 a}{27 x^{2}}$

Step 2.1.2

Factor $9 x^{2}$ out of $27 x^{2}$ .

$\frac{9 x^{2} (x^{3})}{a^{2} - b^{2}} \cdot \frac{3 b - 3 a}{9 x^{2} (3)}$

Step 2.1.3

Cancel the common factor.

$\frac{9 x^{2} x^{3}}{a^{2} - b^{2}} \cdot \frac{3 b - 3 a}{9 x^{2} \cdot 3}$

Step 2.1.4

Rewrite the expression.

$\frac{x^{3}}{a^{2} - b^{2}} \cdot \frac{3 b - 3 a}{3}$

Step 2.2

Multiply $\frac{x^{3}}{a^{2} - b^{2}}$ by $\frac{3 b - 3 a}{3}$ .

$\frac{x^{3} (3 b - 3 a)}{(a^{2} - b^{2}) \cdot 3}$

Step 2.3

Cancel the common factor of $3 b - 3 a$ and $3$ .

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Step 2.3.1

Factor $3$ out of $x^{3} (3 b - 3 a)$ .

$\frac{3 (x^{3} (b - a))}{(a^{2} - b^{2}) \cdot 3}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $3$ out of $(a^{2} - b^{2}) \cdot 3$ .

$\frac{3 (x^{3} (b - a))}{3 \cdot (a^{2} - b^{2})}$

Step 2.3.2.2

Cancel the common factor.

$\frac{3 (x^{3} (b - a))}{3 \cdot (a^{2} - b^{2})}$

Step 2.3.2.3

Rewrite the expression.

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

Step 3

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

$\frac{x^{3} (b - a)}{(a + b) (a - b)}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $b - a$ and $a - b$ .

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Step 4.1.1

Factor $- 1$ out of $b$ .

$\frac{x^{3} (- 1 (- b) - a)}{(a + b) (a - b)}$

Step 4.1.2

Factor $- 1$ out of $- a$ .

$\frac{x^{3} (- 1 (- b) - (a))}{(a + b) (a - b)}$

Step 4.1.3

Factor $- 1$ out of $- 1 (- b) - (a)$ .

$\frac{x^{3} (- 1 (- b + a))}{(a + b) (a - b)}$

Step 4.1.4

Reorder terms.

$\frac{x^{3} (- 1 (a - b))}{(a + b) (a - b)}$

Step 4.1.5

Cancel the common factor.

$\frac{x^{3} (- 1 (a - b))}{(a + b) (a - b)}$

Step 4.1.6

Rewrite the expression.

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

Step 4.2

Simplify the expression.

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Step 4.2.1

Move $- 1$ to the left of $x^{3}$ .

$\frac{- 1 \cdot x^{3}}{a + b}$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{x^{3}}{a + b}$