Simplify ((t+3)/(t+4))÷(t^2+7t+12)

$\frac{t + 3}{t + 4} \div (t^{2} + 7 t + 12)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{t + 3}{t + 4}}{t^{2} + 7 t + 12}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{t + 3}{t + 4} \cdot \frac{1}{t^{2} + 7 t + 12}$

Step 3

Factor $t^{2} + 7 t + 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $7$ .

$3, 4$

Write the factored form using these integers.

$\frac{t + 3}{t + 4} \cdot \frac{1}{(t + 3) (t + 4)}$

Step 4

Simplify terms.

Cancel the common factor of $t + 3$ .

Cancel the common factor.

$\frac{t + 3}{t + 4} \cdot \frac{1}{(t + 3) (t + 4)}$

Rewrite the expression.

$\frac{1}{t + 4} \cdot \frac{1}{t + 4}$

Multiply $\frac{1}{t + 4}$ by $\frac{1}{t + 4}$ .

$\frac{1}{(t + 4) (t + 4)}$

Step 5

Raise $t + 4$ to the power of $1$ .

$\frac{1}{{(t + 4)}^{1} (t + 4)}$

Step 6

Raise $t + 4$ to the power of $1$ .

$\frac{1}{{(t + 4)}^{1} {(t + 4)}^{1}}$

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{(t + 4)}^{1 + 1}}$

Step 8

Add $1$ and $1$ .

$\frac{1}{{(t + 4)}^{2}}$

Simplify ((t+3)/(t+4))÷(t^2+7t+12)

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