Basic Math Examples

\frac{{(t - 9)}^{2}}{25 - t^{2}} \div \frac{t^{2} - 81}{t - 5}

$\frac{{(t - 9)}^{2}}{25 - t^{2}} \div \frac{t^{2} - 81}{t - 5}$

Step 1

To divide by a fraction, multiply by its reciprocal.

\frac{{(t - 9)}^{2}}{25 - t^{2}} \cdot \frac{t - 5}{t^{2} - 81}

$\frac{{(t - 9)}^{2}}{25 - t^{2}} \cdot \frac{t - 5}{t^{2} - 81}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

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

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 5

$a = 5$ and

b = t

$b = t$ .

\frac{{(t - 9)}^{2}}{(5 + t) (5 - t)} \cdot \frac{t - 5}{t^{2} - 81}

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

\frac{{(t - 9)}^{2}}{(5 + t) (5 - t)} \cdot \frac{t - 5}{t^{2} - 81}

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

Step 3

Simplify the denominator.

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Step 3.1

Rewrite

81

$81$ as

9^{2}

$9^{2}$ .

\frac{{(t - 9)}^{2}}{(5 + t) (5 - t)} \cdot \frac{t - 5}{t^{2} - 9^{2}}

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

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = t

$a = t$ and

b = 9

$b = 9$ .

\frac{{(t - 9)}^{2}}{(5 + t) (5 - t)} \cdot \frac{t - 5}{(t + 9) (t - 9)}

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

\frac{{(t - 9)}^{2}}{(5 + t) (5 - t)} \cdot \frac{t - 5}{(t + 9) (t - 9)}

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of

t - 9

$t - 9$ .

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Step 4.1.1

Factor

t - 9

$t - 9$ out of

{(t - 9)}^{2}

${(t - 9)}^{2}$ .

\frac{(t - 9) (t - 9)}{(5 + t) (5 - t)} \cdot \frac{t - 5}{(t + 9) (t - 9)}

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

Step 4.1.2

Factor

t - 9

$t - 9$ out of

(t + 9) (t - 9)

$(t + 9) (t - 9)$ .

\frac{(t - 9) (t - 9)}{(5 + t) (5 - t)} \cdot \frac{t - 5}{(t - 9) (t + 9)}

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

Multiply

$\frac{t - 9}{(5 + t) (5 - t)}$ by

$\frac{t - 5}{t + 9}$ .

$\frac{(t - 9) (t - 5)}{(5 + t) (5 - t) (t + 9)}$

Step 4.3

Cancel the common factor of

$t - 5$ and

$5 - t$ .

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Step 4.3.1

Factor

$- 1$ out of

$t$ .

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

Step 4.3.2

Rewrite

$- 5$ as

$- 1 (5)$ .

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

Step 4.3.3

Factor

$- 1$ out of

$- 1 (- t) - 1 (5)$ .

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

Step 4.3.4

Reorder terms.

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

Step 4.3.5

Cancel the common factor.

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

Step 4.3.6

Rewrite the expression.

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Move

$- 1$ to the left of

$t - 9$ .

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

Step 4.4.2

Move the negative in front of the fraction.

$- \frac{t - 9}{(5 + t) (t + 9)}$