Finite Math Examples

$z = {(1 + i)}^{7}$

Step 1

Subtract ${(1 + i)}^{7}$ from both sides of the equation.

$z - {(1 + i)}^{7} = 0$

Step 2

Simplify each term.

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Step 2.1

Use the Binomial Theorem.

$z - (1^{7} + 7 \cdot 1^{6} i + 21 \cdot 1^{5} i^{2} + 35 \cdot 1^{4} i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

One to any power is one.

$z - (1 + 7 \cdot 1^{6} i + 21 \cdot 1^{5} i^{2} + 35 \cdot 1^{4} i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.2

One to any power is one.

$z - (1 + 7 \cdot 1 i + 21 \cdot 1^{5} i^{2} + 35 \cdot 1^{4} i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.3

Multiply $7$ by $1$ .

$z - (1 + 7 i + 21 \cdot 1^{5} i^{2} + 35 \cdot 1^{4} i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.4

One to any power is one.

$z - (1 + 7 i + 21 \cdot 1 i^{2} + 35 \cdot 1^{4} i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.5

Multiply $21$ by $1$ .

$z - (1 + 7 i + 21 i^{2} + 35 \cdot 1^{4} i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.6

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

$z - (1 + 7 i + 21 \cdot - 1 + 35 \cdot 1^{4} i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.7

Multiply $21$ by $- 1$ .

$z - (1 + 7 i - 21 + 35 \cdot 1^{4} i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.8

One to any power is one.

$z - (1 + 7 i - 21 + 35 \cdot 1 i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.9

Multiply $35$ by $1$ .

$z - (1 + 7 i - 21 + 35 i^{3} + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.10

Factor out $i^{2}$ .

$z - (1 + 7 i - 21 + 35 (i^{2} \cdot i) + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.11

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

$z - (1 + 7 i - 21 + 35 (- 1 \cdot i) + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.12

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

$z - (1 + 7 i - 21 + 35 (- i) + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.13

Multiply $- 1$ by $35$ .

$z - (1 + 7 i - 21 - 35 i + 35 \cdot 1^{3} i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.14

One to any power is one.

$z - (1 + 7 i - 21 - 35 i + 35 \cdot 1 i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.15

Multiply $35$ by $1$ .

$z - (1 + 7 i - 21 - 35 i + 35 i^{4} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.16

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

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Step 2.2.16.1

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

$z - (1 + 7 i - 21 - 35 i + 35 {(i^{2})}^{2} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.16.2

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

$z - (1 + 7 i - 21 - 35 i + 35 {(- 1)}^{2} + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.16.3

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

$z - (1 + 7 i - 21 - 35 i + 35 \cdot 1 + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.17

Multiply $35$ by $1$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 \cdot 1^{2} i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.18

One to any power is one.

$z - (1 + 7 i - 21 - 35 i + 35 + 21 \cdot 1 i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.19

Multiply $21$ by $1$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i^{5} + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.20

Factor out $i^{4}$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 (i^{4} i) + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.21

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

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Step 2.2.21.1

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 ({(i^{2})}^{2} i) + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.21.2

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 ({(- 1)}^{2} i) + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.21.3

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 (1 i) + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.22

Multiply $i$ by $1$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i + 7 \cdot 1 i^{6} + i^{7}) = 0$

Step 2.2.23

Multiply $7$ by $1$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i + 7 i^{6} + i^{7}) = 0$

Step 2.2.24

Factor out $i^{4}$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i + 7 (i^{4} i^{2}) + i^{7}) = 0$

Step 2.2.25

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

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Step 2.2.25.1

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i + 7 ({(i^{2})}^{2} i^{2}) + i^{7}) = 0$

Step 2.2.25.2

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i + 7 ({(- 1)}^{2} i^{2}) + i^{7}) = 0$

Step 2.2.25.3

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i + 7 (1 i^{2}) + i^{7}) = 0$

Step 2.2.26

Multiply $i^{2}$ by $1$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i + 7 i^{2} + i^{7}) = 0$

Step 2.2.27

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i + 7 \cdot - 1 + i^{7}) = 0$

Step 2.2.28

Multiply $7$ by $- 1$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 + i^{7}) = 0$

Step 2.2.29

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

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Step 2.2.29.1

Factor out $i^{4}$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 + i^{4} i^{3}) = 0$

Step 2.2.29.2

Factor out $i^{2}$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 + i^{4} (i^{2} \cdot i)) = 0$

Step 2.2.30

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

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Step 2.2.30.1

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 + {(i^{2})}^{2} (i^{2} \cdot i)) = 0$

Step 2.2.30.2

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 + {(- 1)}^{2} (i^{2} \cdot i)) = 0$

Step 2.2.30.3

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 + 1 (i^{2} \cdot i)) = 0$

Step 2.2.31

Multiply $i^{2} \cdot i$ by $1$ .

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 + i^{2} \cdot i) = 0$

Step 2.2.32

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 - 1 \cdot i) = 0$

Step 2.2.33

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

$z - (1 + 7 i - 21 - 35 i + 35 + 21 i - 7 - i) = 0$

Step 2.3

Subtract $21$ from $1$ .

$z - (- 20 + 7 i - 35 i + 35 + 21 i - 7 - i) = 0$

Step 2.4

Add $- 20$ and $35$ .

$z - (15 + 7 i - 35 i + 21 i - 7 - i) = 0$

Step 2.5

Subtract $7$ from $15$ .

$z - (8 + 7 i - 35 i + 21 i - i) = 0$

Step 2.6

Subtract $35 i$ from $7 i$ .

$z - (8 - 28 i + 21 i - i) = 0$

Step 2.7

Add $- 28 i$ and $21 i$ .

$z - (8 - 7 i - i) = 0$

Step 2.8

Subtract $i$ from $- 7 i$ .

$z - (8 - 8 i) = 0$

Step 2.9

Apply the distributive property.

$z - 1 \cdot 8 - (- 8 i) = 0$

Step 2.10

Multiply $- 1$ by $8$ .

$z - 8 - (- 8 i) = 0$

Step 2.11

Multiply $- 8$ by $- 1$ .

$z - 8 + 8 i = 0$

Step 3

Move all terms not containing $z$ to the right side of the equation.

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Step 3.1

Add $8$ to both sides of the equation.

$z + 8 i = 8$

Step 3.2

Subtract $8 i$ from both sides of the equation.

$z = 8 - 8 i$