Finite Math Examples

${(5 u + 2 v)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(5 u)}^{4 - k} \cdot {(2 v)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} \cdot {(5 u)}^{4 - 0} \cdot {(2 v)}^{0} + \frac{4!}{(4 - 1)! 1!} \cdot {(5 u)}^{4 - 1} \cdot {(2 v)}^{1} + \frac{4!}{(4 - 2)! 2!} \cdot {(5 u)}^{4 - 2} \cdot {(2 v)}^{2} + \frac{4!}{(4 - 3)! 3!} \cdot {(5 u)}^{4 - 3} \cdot {(2 v)}^{3} + \frac{4!}{(4 - 4)! 4!} \cdot {(5 u)}^{4 - 4} \cdot {(2 v)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(5 u)}^{4} \cdot {(2 v)}^{0} + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(5 u)}^{4}$ by $1$ .

${(5 u)}^{4} \cdot {(2 v)}^{0} + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.2

Apply the product rule to $5 u$ .

$5^{4} u^{4} \cdot {(2 v)}^{0} + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.3

Raise $5$ to the power of $4$ .

$625 u^{4} \cdot {(2 v)}^{0} + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.4

Apply the product rule to $2 v$ .

$625 u^{4} \cdot (2^{0} v^{0}) + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.5

Rewrite using the commutative property of multiplication.

$625 \cdot 2^{0} u^{4} v^{0} + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.6

Anything raised to $0$ is $1$ .

$625 \cdot 1 u^{4} v^{0} + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.7

Multiply $625$ by $1$ .

$625 u^{4} v^{0} + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.8

Anything raised to $0$ is $1$ .

$625 u^{4} \cdot 1 + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.9

Multiply $625$ by $1$ .

$625 u^{4} + 4 \cdot {(5 u)}^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.10

Apply the product rule to $5 u$ .

$625 u^{4} + 4 \cdot (5^{3} u^{3}) \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.11

Raise $5$ to the power of $3$ .

$625 u^{4} + 4 \cdot (125 u^{3}) \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.12

Multiply $125$ by $4$ .

$625 u^{4} + 500 u^{3} \cdot {(2 v)}^{1} + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.13

Simplify.

$625 u^{4} + 500 u^{3} \cdot (2 v) + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.14

Rewrite using the commutative property of multiplication.

$625 u^{4} + 500 \cdot 2 u^{3} v + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.15

Multiply $500$ by $2$ .

$625 u^{4} + 1000 u^{3} v + 6 \cdot {(5 u)}^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.16

Apply the product rule to $5 u$ .

$625 u^{4} + 1000 u^{3} v + 6 \cdot (5^{2} u^{2}) \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.17

Raise $5$ to the power of $2$ .

$625 u^{4} + 1000 u^{3} v + 6 \cdot (25 u^{2}) \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.18

Multiply $25$ by $6$ .

$625 u^{4} + 1000 u^{3} v + 150 u^{2} \cdot {(2 v)}^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.19

Apply the product rule to $2 v$ .

$625 u^{4} + 1000 u^{3} v + 150 u^{2} \cdot (2^{2} v^{2}) + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.20

Rewrite using the commutative property of multiplication.

$625 u^{4} + 1000 u^{3} v + 150 \cdot 2^{2} u^{2} v^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.21

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

$625 u^{4} + 1000 u^{3} v + 150 \cdot 4 u^{2} v^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.22

Multiply $150$ by $4$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 4 \cdot {(5 u)}^{1} \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.23

Simplify.

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 4 \cdot (5 u) \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.24

Multiply $5$ by $4$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 20 u \cdot {(2 v)}^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.25

Apply the product rule to $2 v$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 20 u \cdot (2^{3} v^{3}) + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.26

Rewrite using the commutative property of multiplication.

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 20 \cdot 2^{3} u v^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.27

Raise $2$ to the power of $3$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 20 \cdot 8 u v^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.28

Multiply $20$ by $8$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + 1 \cdot {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.29

Multiply ${(5 u)}^{0}$ by $1$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + {(5 u)}^{0} \cdot {(2 v)}^{4}$

Step 4.30

Apply the product rule to $5 u$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + 5^{0} u^{0} \cdot {(2 v)}^{4}$

Step 4.31

Anything raised to $0$ is $1$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + 1 u^{0} \cdot {(2 v)}^{4}$

Step 4.32

Multiply $u^{0}$ by $1$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + u^{0} \cdot {(2 v)}^{4}$

Step 4.33

Anything raised to $0$ is $1$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + 1 \cdot {(2 v)}^{4}$

Step 4.34

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

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + {(2 v)}^{4}$

Step 4.35

Apply the product rule to $2 v$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + 2^{4} v^{4}$

Step 4.36

Raise $2$ to the power of $4$ .

$625 u^{4} + 1000 u^{3} v + 600 u^{2} v^{2} + 160 u v^{3} + 16 v^{4}$