Precalculus Examples

$\frac{t - 3}{t^{2} - 9} \cdot \frac{t + 3}{t^{2} + 9}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{t - 3}{t^{2} - 3^{2}} \cdot \frac{t + 3}{t^{2} + 9}$

Step 1.2

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 $b = 3$ .

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $t + 3$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

$\frac{t - 3}{t - 3} \cdot \frac{1}{t^{2} + 9}$

Step 2.2

Multiply $\frac{t - 3}{t - 3}$ by $\frac{1}{t^{2} + 9}$ .

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

Step 2.3

Cancel the common factor of $t - 3$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

$\frac{1}{t^{2} + 9}$