Basic Math Examples

$\frac{81 b^{2} - 25}{b^{2} - 1} \div \frac{9 b - 5}{b - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{81 b^{2} - 25}{b^{2} - 1} \cdot \frac{b - 1}{9 b - 5}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(9 b)}^{2} - 25}{b^{2} - 1} \cdot \frac{b - 1}{9 b - 5}$

Step 2.2

Rewrite $25$ as $5^{2}$ .

$\frac{{(9 b)}^{2} - 5^{2}}{b^{2} - 1} \cdot \frac{b - 1}{9 b - 5}$

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 $b = 5$ .

$\frac{(9 b + 5) (9 b - 5)}{b^{2} - 1} \cdot \frac{b - 1}{9 b - 5}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$\frac{(9 b + 5) (9 b - 5)}{b^{2} - 1^{2}} \cdot \frac{b - 1}{9 b - 5}$

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 $b = 1$ .

$\frac{(9 b + 5) (9 b - 5)}{(b + 1) (b - 1)} \cdot \frac{b - 1}{9 b - 5}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $9 b - 5$ .

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Step 4.1.1

Factor $9 b - 5$ out of $(9 b + 5) (9 b - 5)$ .

$\frac{(9 b - 5) (9 b + 5)}{(b + 1) (b - 1)} \cdot \frac{b - 1}{9 b - 5}$

Step 4.1.2

Cancel the common factor.

$\frac{(9 b - 5) (9 b + 5)}{(b + 1) (b - 1)} \cdot \frac{b - 1}{9 b - 5}$

Step 4.1.3

Rewrite the expression.

$\frac{9 b + 5}{(b + 1) (b - 1)} (b - 1)$

Step 4.2

Cancel the common factor of $b - 1$ .

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Step 4.2.1

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

$\frac{9 b + 5}{(b - 1) (b + 1)} (b - 1)$

Step 4.2.2

Cancel the common factor.

$\frac{9 b + 5}{(b - 1) (b + 1)} (b - 1)$

Step 4.2.3

Rewrite the expression.

$\frac{9 b + 5}{b + 1}$