Precalculus Examples

${(4 p + 2)}^{4}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(4 p)}^{4} \cdot {(2)}^{0} + 4 \cdot {(4 p)}^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(4 p)}^{4}$ by $1$ .

${(4 p)}^{4} \cdot {(2)}^{0} + 4 \cdot {(4 p)}^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.2

Apply the product rule to $4 p$ .

$4^{4} p^{4} \cdot {(2)}^{0} + 4 \cdot {(4 p)}^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.3

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

$256 p^{4} \cdot {(2)}^{0} + 4 \cdot {(4 p)}^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.4

Anything raised to $0$ is $1$ .

$256 p^{4} \cdot 1 + 4 \cdot {(4 p)}^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.5

Multiply $256$ by $1$ .

$256 p^{4} + 4 \cdot {(4 p)}^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.6

Apply the product rule to $4 p$ .

$256 p^{4} + 4 \cdot (4^{3} p^{3}) \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.7

Multiply $4$ by $4^{3}$ by adding the exponents.

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Step 4.7.1

Move $4^{3}$ .

$256 p^{4} + 4^{3} \cdot 4 \cdot p^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.7.2

Multiply $4^{3}$ by $4$ .

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Step 4.7.2.1

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

$256 p^{4} + 4^{3} \cdot 4^{1} \cdot p^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.7.2.2

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

$256 p^{4} + 4^{3 + 1} \cdot p^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.7.3

Add $3$ and $1$ .

$256 p^{4} + 4^{4} \cdot p^{3} \cdot {(2)}^{1} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.8

Simplify $4^{4} \cdot p^{3} \cdot {(2)}^{1}$ .

$256 p^{4} + 4^{4} \cdot p^{3} \cdot 2 + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.9

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

$256 p^{4} + 256 \cdot p^{3} \cdot 2 + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.10

Multiply $2$ by $256$ .

$256 p^{4} + 512 p^{3} + 6 \cdot {(4 p)}^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.11

Apply the product rule to $4 p$ .

$256 p^{4} + 512 p^{3} + 6 \cdot (4^{2} p^{2}) \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.12

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

$256 p^{4} + 512 p^{3} + 6 \cdot (16 p^{2}) \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.13

Multiply $16$ by $6$ .

$256 p^{4} + 512 p^{3} + 96 p^{2} \cdot {(2)}^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.14

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

$256 p^{4} + 512 p^{3} + 96 p^{2} \cdot 4 + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.15

Multiply $4$ by $96$ .

$256 p^{4} + 512 p^{3} + 384 p^{2} + 4 \cdot {(4 p)}^{1} \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.16

Simplify.

$256 p^{4} + 512 p^{3} + 384 p^{2} + 4 \cdot (4 p) \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.17

Multiply $4$ by $4$ .

$256 p^{4} + 512 p^{3} + 384 p^{2} + 16 p \cdot {(2)}^{3} + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.18

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

$256 p^{4} + 512 p^{3} + 384 p^{2} + 16 p \cdot 8 + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.19

Multiply $8$ by $16$ .

$256 p^{4} + 512 p^{3} + 384 p^{2} + 128 p + 1 \cdot {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.20

Multiply ${(4 p)}^{0}$ by $1$ .

$256 p^{4} + 512 p^{3} + 384 p^{2} + 128 p + {(4 p)}^{0} \cdot {(2)}^{4}$

Step 4.21

Apply the product rule to $4 p$ .

$256 p^{4} + 512 p^{3} + 384 p^{2} + 128 p + 4^{0} p^{0} \cdot {(2)}^{4}$

Step 4.22

Anything raised to $0$ is $1$ .

$256 p^{4} + 512 p^{3} + 384 p^{2} + 128 p + 1 p^{0} \cdot {(2)}^{4}$

Step 4.23

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

$256 p^{4} + 512 p^{3} + 384 p^{2} + 128 p + p^{0} \cdot {(2)}^{4}$

Step 4.24

Anything raised to $0$ is $1$ .

$256 p^{4} + 512 p^{3} + 384 p^{2} + 128 p + 1 \cdot {(2)}^{4}$

Step 4.25

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

$256 p^{4} + 512 p^{3} + 384 p^{2} + 128 p + {(2)}^{4}$

Step 4.26

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

$256 p^{4} + 512 p^{3} + 384 p^{2} + 128 p + 16$