Precalculus Examples

${(r^{2} - s)}^{4}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(r^{2})}^{4 - k} \cdot {(- s)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} \cdot {(r^{2})}^{4 - 0} \cdot {(- s)}^{0} + \frac{4!}{(4 - 1)! 1!} \cdot {(r^{2})}^{4 - 1} \cdot {(- s)}^{1} + \frac{4!}{(4 - 2)! 2!} \cdot {(r^{2})}^{4 - 2} \cdot {(- s)}^{2} + \frac{4!}{(4 - 3)! 3!} \cdot {(r^{2})}^{4 - 3} \cdot {(- s)}^{3} + \frac{4!}{(4 - 4)! 4!} \cdot {(r^{2})}^{4 - 4} \cdot {(- s)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(r^{2})}^{4} \cdot {(- s)}^{0} + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(r^{2})}^{4}$ by $1$ .

${(r^{2})}^{4} \cdot {(- s)}^{0} + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.2

Multiply the exponents in ${(r^{2})}^{4}$ .

Tap for more steps...

Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r^{2 \cdot 4} \cdot {(- s)}^{0} + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.2.2

Multiply $2$ by $4$ .

$r^{8} \cdot {(- s)}^{0} + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.3

Apply the product rule to $- s$ .

$r^{8} \cdot ({(- 1)}^{0} s^{0}) + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.4

Rewrite using the commutative property of multiplication.

${(- 1)}^{0} r^{8} s^{0} + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.5

Anything raised to $0$ is $1$ .

$1 r^{8} s^{0} + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.6

Multiply $r^{8}$ by $1$ .

$r^{8} s^{0} + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.7

Anything raised to $0$ is $1$ .

$r^{8} \cdot 1 + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.8

Multiply $r^{8}$ by $1$ .

$r^{8} + 4 \cdot {(r^{2})}^{3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.9

Multiply the exponents in ${(r^{2})}^{3}$ .

Tap for more steps...

Step 4.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r^{8} + 4 \cdot r^{2 \cdot 3} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.9.2

Multiply $2$ by $3$ .

$r^{8} + 4 \cdot r^{6} \cdot {(- s)}^{1} + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.10

Simplify.

$r^{8} + 4 r^{6} \cdot (- s) + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.11

Rewrite using the commutative property of multiplication.

$r^{8} + 4 \cdot - 1 r^{6} s + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.12

Multiply $4$ by $- 1$ .

$r^{8} - 4 r^{6} s + 6 \cdot {(r^{2})}^{2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.13

Multiply the exponents in ${(r^{2})}^{2}$ .

Tap for more steps...

Step 4.13.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r^{8} - 4 r^{6} s + 6 \cdot r^{2 \cdot 2} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.13.2

Multiply $2$ by $2$ .

$r^{8} - 4 r^{6} s + 6 \cdot r^{4} \cdot {(- s)}^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.14

Apply the product rule to $- s$ .

$r^{8} - 4 r^{6} s + 6 r^{4} \cdot ({(- 1)}^{2} s^{2}) + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.15

Rewrite using the commutative property of multiplication.

$r^{8} - 4 r^{6} s + 6 \cdot {(- 1)}^{2} r^{4} s^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.16

Raise $- 1$ to the power of $2$ .

$r^{8} - 4 r^{6} s + 6 \cdot 1 r^{4} s^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.17

Multiply $6$ by $1$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} + 4 \cdot {(r^{2})}^{1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.18

Multiply the exponents in ${(r^{2})}^{1}$ .

Tap for more steps...

Step 4.18.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} + 4 \cdot r^{2 \cdot 1} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.18.2

Multiply $2$ by $1$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} + 4 \cdot r^{2} \cdot {(- s)}^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.19

Apply the product rule to $- s$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} + 4 r^{2} \cdot ({(- 1)}^{3} s^{3}) + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.20

Rewrite using the commutative property of multiplication.

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} + 4 \cdot {(- 1)}^{3} r^{2} s^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.21

Raise $- 1$ to the power of $3$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} + 4 \cdot - 1 r^{2} s^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.22

Multiply $4$ by $- 1$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + 1 \cdot {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.23

Multiply ${(r^{2})}^{0}$ by $1$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + {(r^{2})}^{0} \cdot {(- s)}^{4}$

Step 4.24

Multiply the exponents in ${(r^{2})}^{0}$ .

Tap for more steps...

Step 4.24.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + r^{2 \cdot 0} \cdot {(- s)}^{4}$

Step 4.24.2

Multiply $2$ by $0$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + r^{0} \cdot {(- s)}^{4}$

Step 4.25

Anything raised to $0$ is $1$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + 1 \cdot {(- s)}^{4}$

Step 4.26

Multiply ${(- s)}^{4}$ by $1$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + {(- s)}^{4}$

Step 4.27

Apply the product rule to $- s$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + {(- 1)}^{4} s^{4}$

Step 4.28

Raise $- 1$ to the power of $4$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + 1 s^{4}$

Step 4.29

Multiply $s^{4}$ by $1$ .

$r^{8} - 4 r^{6} s + 6 r^{4} s^{2} - 4 r^{2} s^{3} + s^{4}$