Simplify ((c+3)/(c^2-4))÷((c+2)/(3(c^2-9)))

$\frac{c + 3}{c^{2} - 4} \div \frac{c + 2}{3 (c^{2} - 9)}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{c + 3}{c^{2} - 4} \cdot \frac{3 (c^{2} - 9)}{c + 2}$

Step 2

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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

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

Step 3

Simplify the numerator.

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = 3$ .

$\frac{c + 3}{(c + 2) (c - 2)} \cdot \frac{3 (c + 3) (c - 3)}{c + 2}$

Step 4

Multiply $\frac{c + 3}{(c + 2) (c - 2)} \cdot \frac{3 (c + 3) (c - 3)}{c + 2}$ .

Multiply $\frac{c + 3}{(c + 2) (c - 2)}$ by $\frac{3 (c + 3) (c - 3)}{c + 2}$ .

$\frac{(c + 3) (3 (c + 3) (c - 3))}{(c + 2) (c - 2) (c + 2)}$

Raise $c + 3$ to the power of $1$ .

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

Raise $c + 3$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Raise $c + 2$ to the power of $1$ .

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

Raise $c + 2$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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