Algebra Examples

${(5 + 5 i)}^{4}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify the polynomial result.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Apply the product rule to $5 i$ .

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

Step 4.1.4

Anything raised to $0$ is $1$ .

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

Step 4.1.5

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

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

Step 4.1.6

Anything raised to $0$ is $1$ .

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

Step 4.1.7

Multiply $625$ by $1$ .

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

Step 4.1.8

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

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

Step 4.1.9

Multiply $4$ by $125$ .

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

Step 4.1.10

Simplify.

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

Step 4.1.11

Multiply $5$ by $500$ .

$625 + 2500 i + 6 \cdot {(5)}^{2} \cdot {(5 i)}^{2} + 4 \cdot {(5)}^{1} \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.12

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

$625 + 2500 i + 6 \cdot 25 \cdot {(5 i)}^{2} + 4 \cdot {(5)}^{1} \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.13

Multiply $6$ by $25$ .

$625 + 2500 i + 150 \cdot {(5 i)}^{2} + 4 \cdot {(5)}^{1} \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.14

Apply the product rule to $5 i$ .

$625 + 2500 i + 150 \cdot (5^{2} i^{2}) + 4 \cdot {(5)}^{1} \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.15

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

$625 + 2500 i + 150 \cdot (25 i^{2}) + 4 \cdot {(5)}^{1} \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.16

Rewrite $i^{2}$ as $- 1$ .

$625 + 2500 i + 150 \cdot (25 \cdot - 1) + 4 \cdot {(5)}^{1} \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.17

Multiply $150 (25 \cdot - 1)$ .

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Step 4.1.17.1

Multiply $25$ by $- 1$ .

$625 + 2500 i + 150 \cdot - 25 + 4 \cdot {(5)}^{1} \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.17.2

Multiply $150$ by $- 25$ .

$625 + 2500 i - 3750 + 4 \cdot {(5)}^{1} \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.18

Evaluate the exponent.

$625 + 2500 i - 3750 + 4 \cdot 5 \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.19

Multiply $4$ by $5$ .

$625 + 2500 i - 3750 + 20 \cdot {(5 i)}^{3} + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.20

Apply the product rule to $5 i$ .

$625 + 2500 i - 3750 + 20 \cdot (5^{3} i^{3}) + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.21

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

$625 + 2500 i - 3750 + 20 \cdot (125 i^{3}) + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.22

Factor out $i^{2}$ .

$625 + 2500 i - 3750 + 20 \cdot (125 (i^{2} \cdot i)) + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.23

Rewrite $i^{2}$ as $- 1$ .

$625 + 2500 i - 3750 + 20 \cdot (125 (- 1 \cdot i)) + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.24

Rewrite $- 1 i$ as $- i$ .

$625 + 2500 i - 3750 + 20 \cdot (125 (- i)) + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.25

Multiply $- 1$ by $125$ .

$625 + 2500 i - 3750 + 20 \cdot (- 125 i) + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.26

Multiply $- 125$ by $20$ .

$625 + 2500 i - 3750 - 2500 i + 1 \cdot {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.27

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

$625 + 2500 i - 3750 - 2500 i + {(5)}^{0} \cdot {(5 i)}^{4}$

Step 4.1.28

Anything raised to $0$ is $1$ .

$625 + 2500 i - 3750 - 2500 i + 1 \cdot {(5 i)}^{4}$

Step 4.1.29

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

$625 + 2500 i - 3750 - 2500 i + {(5 i)}^{4}$

Step 4.1.30

Apply the product rule to $5 i$ .

$625 + 2500 i - 3750 - 2500 i + 5^{4} i^{4}$

Step 4.1.31

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

$625 + 2500 i - 3750 - 2500 i + 625 i^{4}$

Step 4.1.32

Rewrite $i^{4}$ as $1$ .

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Step 4.1.32.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$625 + 2500 i - 3750 - 2500 i + 625 {(i^{2})}^{2}$

Step 4.1.32.2

Rewrite $i^{2}$ as $- 1$ .

$625 + 2500 i - 3750 - 2500 i + 625 {(- 1)}^{2}$

Step 4.1.32.3

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

$625 + 2500 i - 3750 - 2500 i + 625 \cdot 1$

Step 4.1.33

Multiply $625$ by $1$ .

$625 + 2500 i - 3750 - 2500 i + 625$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract $3750$ from $625$ .

$- 3125 + 2500 i - 2500 i + 625$

Step 4.2.2

Add $- 3125$ and $625$ .

$- 2500 + 2500 i - 2500 i$

Step 4.2.3

Subtract $2500 i$ from $2500 i$ .

$- 2500 + 0$

Step 4.2.4

Add $- 2500$ and $0$ .

$- 2500$