Simplify r+(9/((r-2)(r+10)))÷((10r(r+8))/((r-2)(r+10)))

$r + \frac{9}{(r - 2) (r + 10)} \div \frac{10 r (r + 8)}{(r - 2) (r + 10)}$

Step 1

Simplify each term.

To divide by a fraction, multiply by its reciprocal.

$r + \frac{9}{(r - 2) (r + 10)} \cdot \frac{(r - 2) (r + 10)}{10 r (r + 8)}$

Cancel the common factor of $(r - 2) (r + 10)$ .

Cancel the common factor.

$r + \frac{9}{(r - 2) (r + 10)} \cdot \frac{(r - 2) (r + 10)}{10 r (r + 8)}$

Rewrite the expression.

$r + 9 \frac{1}{10 r (r + 8)}$

Combine $9$ and $\frac{1}{10 r (r + 8)}$ .

$r + \frac{9}{10 r (r + 8)}$

Step 2

To write $r$ as a fraction with a common denominator, multiply by $\frac{10 r (r + 8)}{10 r (r + 8)}$ .

$r \cdot \frac{10 r (r + 8)}{10 r (r + 8)} + \frac{9}{10 r (r + 8)}$

Step 3

Simplify terms.

Combine $r$ and $\frac{10 r (r + 8)}{10 r (r + 8)}$ .

$\frac{r (10 r (r + 8))}{10 r (r + 8)} + \frac{9}{10 r (r + 8)}$

Combine the numerators over the common denominator.

$\frac{r (10 r (r + 8)) + 9}{10 r (r + 8)}$

Step 4

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$\frac{10 r (r (r + 8)) + 9}{10 r (r + 8)}$

Multiply $r$ by $r$ by adding the exponents.

Move $r$ .

$\frac{10 (r \cdot r) (r + 8) + 9}{10 r (r + 8)}$

Multiply $r$ by $r$ .

$\frac{10 r^{2} (r + 8) + 9}{10 r (r + 8)}$

Apply the distributive property.

$\frac{10 r^{2} r + 10 r^{2} \cdot 8 + 9}{10 r (r + 8)}$

Multiply $r^{2}$ by $r$ by adding the exponents.

Move $r$ .

$\frac{10 (r \cdot r^{2}) + 10 r^{2} \cdot 8 + 9}{10 r (r + 8)}$

Multiply $r$ by $r^{2}$ .

Raise $r$ to the power of $1$ .

$\frac{10 (r^{1} r^{2}) + 10 r^{2} \cdot 8 + 9}{10 r (r + 8)}$

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

$\frac{10 r^{1 + 2} + 10 r^{2} \cdot 8 + 9}{10 r (r + 8)}$

Add $1$ and $2$ .

$\frac{10 r^{3} + 10 r^{2} \cdot 8 + 9}{10 r (r + 8)}$

Multiply $8$ by $10$ .

$\frac{10 r^{3} + 80 r^{2} + 9}{10 r (r + 8)}$