Basic Math Examples

$\frac{\frac{t^{2} - 1}{9 t + 9}}{\frac{3 t^{2} - 6 t + 3}{27 t + 27}}$

Step 1

Cancel the common factor of $3 t^{2} - 6 t + 3$ and $27 t + 27$ .

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Step 1.1

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

$\frac{\frac{t^{2} - 1}{9 t + 9}}{\frac{3 (t^{2}) - 6 t + 3}{27 t + 27}}$

Step 1.2

Factor $3$ out of $- 6 t$ .

$\frac{\frac{t^{2} - 1}{9 t + 9}}{\frac{3 (t^{2}) + 3 (- 2 t) + 3}{27 t + 27}}$

Step 1.3

Factor $3$ out of $3 (t^{2}) + 3 (- 2 t)$ .

$\frac{\frac{t^{2} - 1}{9 t + 9}}{\frac{3 (t^{2} - 2 t) + 3}{27 t + 27}}$

Step 1.4

Factor $3$ out of $3$ .

$\frac{\frac{t^{2} - 1}{9 t + 9}}{\frac{3 (t^{2} - 2 t) + 3 (1)}{27 t + 27}}$

Step 1.5

Factor $3$ out of $3 (t^{2} - 2 t) + 3 (1)$ .

$\frac{\frac{t^{2} - 1}{9 t + 9}}{\frac{3 (t^{2} - 2 t + 1)}{27 t + 27}}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $3$ out of $27 t$ .

$\frac{\frac{t^{2} - 1}{9 t + 9}}{\frac{3 (t^{2} - 2 t + 1)}{3 (9 t) + 27}}$

Step 1.6.2

Factor $3$ out of $27$ .

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

Step 1.6.3

Factor $3$ out of $3 (9 t) + 3 (9)$ .

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

Step 1.6.4

Cancel the common factor.

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

Step 1.6.5

Rewrite the expression.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Cancel the common factor of $9 t + 9$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$(t^{2} - 1) \frac{1}{t^{2} - 2 t + 1}$

Step 4

Factor using the perfect square rule.

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Step 4.1

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

$(t^{2} - 1) \frac{1}{t^{2} - 2 t + 1^{2}}$

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 t = 2 \cdot t \cdot 1$

Step 4.3

Rewrite the polynomial.

$(t^{2} - 1) \frac{1}{t^{2} - 2 \cdot t \cdot 1 + 1^{2}}$

Step 4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = t$ and $b = 1$ .

$(t^{2} - 1) \frac{1}{{(t - 1)}^{2}}$

Step 5

Multiply $t^{2} - 1$ by $\frac{1}{{(t - 1)}^{2}}$ .

$\frac{t^{2} - 1}{{(t - 1)}^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{t^{2} - 1^{2}}{{(t - 1)}^{2}}$

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 $b = 1$ .

$\frac{(t + 1) (t - 1)}{{(t - 1)}^{2}}$

Step 7

Cancel the common factor of $t - 1$ and ${(t - 1)}^{2}$ .

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Step 7.1

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

$\frac{(t - 1) (t + 1)}{{(t - 1)}^{2}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $t - 1$ out of ${(t - 1)}^{2}$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$\frac{t + 1}{t - 1}$