Basic Math Examples

$\frac{81 b^{2} - 4}{b^{2} - 9} \div \frac{9 b - 2}{b - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{81 b^{2} - 4}{b^{2} - 9} \cdot \frac{b - 3}{9 b - 2}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $81 b^{2}$ as ${(9 b)}^{2}$ .

$\frac{{(9 b)}^{2} - 4}{b^{2} - 9} \cdot \frac{b - 3}{9 b - 2}$

Step 2.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9 b$ and $b = 2$ .

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

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

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

Step 3.2

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

$\frac{(9 b + 2) (9 b - 2)}{(b + 3) (b - 3)} \cdot \frac{b - 3}{9 b - 2}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $9 b - 2$ .

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Step 4.1.1

Factor $9 b - 2$ out of $(9 b + 2) (9 b - 2)$ .

$\frac{(9 b - 2) (9 b + 2)}{(b + 3) (b - 3)} \cdot \frac{b - 3}{9 b - 2}$

Step 4.1.2

Cancel the common factor.

$\frac{(9 b - 2) (9 b + 2)}{(b + 3) (b - 3)} \cdot \frac{b - 3}{9 b - 2}$

Step 4.1.3

Rewrite the expression.

$\frac{9 b + 2}{(b + 3) (b - 3)} (b - 3)$

Step 4.2

Cancel the common factor of $b - 3$ .

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Step 4.2.1

Factor $b - 3$ out of $(b + 3) (b - 3)$ .

$\frac{9 b + 2}{(b - 3) (b + 3)} (b - 3)$

Step 4.2.2

Cancel the common factor.

$\frac{9 b + 2}{(b - 3) (b + 3)} (b - 3)$

Step 4.2.3

Rewrite the expression.

$\frac{9 b + 2}{b + 3}$