Simplify (3(r+s)^3)/((r+s)^-2)

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

Step 1

Move ${(r + s)}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$3 {(r + s)}^{3} {(r + s)}^{2}$

Step 2

Multiply ${(r + s)}^{3}$ by ${(r + s)}^{2}$ by adding the exponents.

Move ${(r + s)}^{2}$ .

$3 ({(r + s)}^{2} {(r + s)}^{3})$

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

$3 {(r + s)}^{2 + 3}$

Add $2$ and $3$ .

$3 {(r + s)}^{5}$

Step 3

Use the Binomial Theorem.

$3 (r^{5} + 5 r^{4} s + 10 r^{3} s^{2} + 10 r^{2} s^{3} + 5 r s^{4} + s^{5})$

Step 4

Apply the distributive property.

$3 r^{5} + 3 (5 r^{4} s) + 3 (10 r^{3} s^{2}) + 3 (10 r^{2} s^{3}) + 3 (5 r s^{4}) + 3 s^{5}$

Step 5

Simplify.

Multiply $5$ by $3$ .

$3 r^{5} + 15 (r^{4} s) + 3 (10 r^{3} s^{2}) + 3 (10 r^{2} s^{3}) + 3 (5 r s^{4}) + 3 s^{5}$

Multiply $10$ by $3$ .

$3 r^{5} + 15 (r^{4} s) + 30 (r^{3} s^{2}) + 3 (10 r^{2} s^{3}) + 3 (5 r s^{4}) + 3 s^{5}$

Multiply $10$ by $3$ .

$3 r^{5} + 15 (r^{4} s) + 30 (r^{3} s^{2}) + 30 (r^{2} s^{3}) + 3 (5 r s^{4}) + 3 s^{5}$

Multiply $5$ by $3$ .

$3 r^{5} + 15 (r^{4} s) + 30 (r^{3} s^{2}) + 30 (r^{2} s^{3}) + 15 (r s^{4}) + 3 s^{5}$

Step 6

Remove parentheses.

$3 r^{5} + 15 r^{4} s + 30 r^{3} s^{2} + 30 r^{2} s^{3} + 15 r s^{4} + 3 s^{5}$