Pre-Algebra Examples

$\frac{3 x^{2} - 27 y^{2}}{30} \div \frac{{(x - 3 y)}^{2}}{6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} - 27 y^{2}}{30} \cdot \frac{6}{{(x - 3 y)}^{2}}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $6$ .

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Step 2.1.1

Factor $6$ out of $30$ .

$\frac{3 x^{2} - 27 y^{2}}{6 (5)} \cdot \frac{6}{{(x - 3 y)}^{2}}$

Step 2.1.2

Cancel the common factor.

$\frac{3 x^{2} - 27 y^{2}}{6 \cdot 5} \cdot \frac{6}{{(x - 3 y)}^{2}}$

Step 2.1.3

Rewrite the expression.

$\frac{3 x^{2} - 27 y^{2}}{5} \cdot \frac{1}{{(x - 3 y)}^{2}}$

Step 2.2

Multiply $\frac{3 x^{2} - 27 y^{2}}{5}$ by $\frac{1}{{(x - 3 y)}^{2}}$ .

$\frac{3 x^{2} - 27 y^{2}}{5 {(x - 3 y)}^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $3$ out of $3 x^{2} - 27 y^{2}$ .

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Step 3.1.1

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

$\frac{3 (x^{2}) - 27 y^{2}}{5 {(x - 3 y)}^{2}}$

Step 3.1.2

Factor $3$ out of $- 27 y^{2}$ .

$\frac{3 (x^{2}) + 3 (- 9 y^{2})}{5 {(x - 3 y)}^{2}}$

Step 3.1.3

Factor $3$ out of $3 (x^{2}) + 3 (- 9 y^{2})$ .

$\frac{3 (x^{2} - 9 y^{2})}{5 {(x - 3 y)}^{2}}$

Step 3.2

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

$\frac{3 (x^{2} - {(3 y)}^{2})}{5 {(x - 3 y)}^{2}}$

Step 3.3

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

$\frac{3 ((x + 3 y) (x - (3 y)))}{5 {(x - 3 y)}^{2}}$

Step 3.4

Multiply $3$ by $- 1$ .

$\frac{3 (x + 3 y) (x - 3 y)}{5 {(x - 3 y)}^{2}}$

Step 4

Cancel the common factor of $x - 3 y$ and ${(x - 3 y)}^{2}$ .

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Step 4.1

Factor $x - 3 y$ out of $3 (x + 3 y) (x - 3 y)$ .

$\frac{(x - 3 y) (3 (x + 3 y))}{5 {(x - 3 y)}^{2}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $x - 3 y$ out of $5 {(x - 3 y)}^{2}$ .

$\frac{(x - 3 y) (3 (x + 3 y))}{(x - 3 y) (5 (x - 3 y))}$

Step 4.2.2

Cancel the common factor.

$\frac{(x - 3 y) (3 (x + 3 y))}{(x - 3 y) (5 (x - 3 y))}$

Step 4.2.3

Rewrite the expression.

$\frac{3 (x + 3 y)}{5 (x - 3 y)}$