Basic Math Examples

$\frac{3 s^{2}}{5 (t^{2} - 81)} \div \frac{2 s}{9 t - t^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 s^{2}}{5 (t^{2} - 81)} \cdot \frac{9 t - t^{2}}{2 s}$

Step 2

Cancel the common factor of $s$ .

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Step 2.1

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

$\frac{s (3 s)}{5 (t^{2} - 81)} \cdot \frac{9 t - t^{2}}{2 s}$

Step 2.2

Factor $s$ out of $2 s$ .

$\frac{s (3 s)}{5 (t^{2} - 81)} \cdot \frac{9 t - t^{2}}{s \cdot 2}$

Step 2.3

Cancel the common factor.

$\frac{s (3 s)}{5 (t^{2} - 81)} \cdot \frac{9 t - t^{2}}{s \cdot 2}$

Step 2.4

Rewrite the expression.

$\frac{3 s}{5 (t^{2} - 81)} \cdot \frac{9 t - t^{2}}{2}$

Step 3

Multiply $\frac{3 s}{5 (t^{2} - 81)}$ by $\frac{9 t - t^{2}}{2}$ .

$\frac{3 s (9 t - t^{2})}{5 (t^{2} - 81) \cdot 2}$

Step 4

Multiply $2$ by $5$ .

$\frac{3 s (9 t - t^{2})}{10 (t^{2} - 81)}$

Step 5

Factor $t$ out of $9 t - t^{2}$ .

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Step 5.1

Factor $t$ out of $9 t$ .

$\frac{3 s (t \cdot 9 - t^{2})}{10 (t^{2} - 81)}$

Step 5.2

Factor $t$ out of $- t^{2}$ .

$\frac{3 s (t \cdot 9 + t (- t))}{10 (t^{2} - 81)}$

Step 5.3

Factor $t$ out of $t \cdot 9 + t (- t)$ .

$\frac{3 s (t (9 - t))}{10 (t^{2} - 81)}$

$\frac{3 s t (9 - t)}{10 (t^{2} - 81)}$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite $81$ as $9^{2}$ .

$\frac{3 s t (9 - t)}{10 (t^{2} - 9^{2})}$

Step 6.2

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

$\frac{3 s t (9 - t)}{10 (t + 9) (t - 9)}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

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

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Step 7.1.1

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{3 s t (- 1 (- 9) - t)}{10 (t + 9) (t - 9)}$

Step 7.1.2

Factor $- 1$ out of $- t$ .

$\frac{3 s t (- 1 (- 9) - (t))}{10 (t + 9) (t - 9)}$

Step 7.1.3

Factor $- 1$ out of $- 1 (- 9) - (t)$ .

$\frac{3 s t (- 1 (- 9 + t))}{10 (t + 9) (t - 9)}$

Step 7.1.4

Reorder terms.

$\frac{3 s t (- 1 (t - 9))}{10 (t + 9) (t - 9)}$

Step 7.1.5

Cancel the common factor.

$\frac{3 s t (- 1 (t - 9))}{10 (t + 9) (t - 9)}$

Step 7.1.6

Rewrite the expression.

$\frac{3 s t \cdot (- 1)}{10 (t + 9)}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Multiply $- 1$ by $3$ .

$\frac{- 3 s t}{10 (t + 9)}$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{3 s t}{10 (t + 9)}$