Algebra Examples

${(5 + 5 i)}^{7}$

Step 1

Use the Binomial Theorem.

$5^{7} + 7 \cdot 5^{6} (5 i) + 21 \cdot 5^{5} {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $5$ to the power of $7$ .

$78125 + 7 \cdot 5^{6} (5 i) + 21 \cdot 5^{5} {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.2

Multiply $5^{6}$ by $5$ by adding the exponents.

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Step 2.1.2.1

Move $5$ .

$78125 + 7 \cdot (5 \cdot 5^{6}) i + 21 \cdot 5^{5} {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.2.2

Multiply $5$ by $5^{6}$ .

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Step 2.1.2.2.1

Raise $5$ to the power of $1$ .

$78125 + 7 \cdot (5^{1} \cdot 5^{6}) i + 21 \cdot 5^{5} {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.2.2.2

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

$78125 + 7 \cdot 5^{1 + 6} i + 21 \cdot 5^{5} {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.2.3

Add $1$ and $6$ .

$78125 + 7 \cdot 5^{7} i + 21 \cdot 5^{5} {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.3

Raise $5$ to the power of $7$ .

$78125 + 7 \cdot 78125 i + 21 \cdot 5^{5} {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.4

Multiply $7$ by $78125$ .

$78125 + 546875 i + 21 \cdot 5^{5} {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.5

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

$78125 + 546875 i + 21 \cdot 3125 {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.6

Multiply $21$ by $3125$ .

$78125 + 546875 i + 65625 {(5 i)}^{2} + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.7

Apply the product rule to $5 i$ .

$78125 + 546875 i + 65625 (5^{2} i^{2}) + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.8

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

$78125 + 546875 i + 65625 (25 i^{2}) + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.9

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

$78125 + 546875 i + 65625 (25 \cdot - 1) + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.10

Multiply $65625 (25 \cdot - 1)$ .

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Step 2.1.10.1

Multiply $25$ by $- 1$ .

$78125 + 546875 i + 65625 \cdot - 25 + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.10.2

Multiply $65625$ by $- 25$ .

$78125 + 546875 i - 1640625 + 35 \cdot 5^{4} {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.11

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

$78125 + 546875 i - 1640625 + 35 \cdot 625 {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.12

Multiply $35$ by $625$ .

$78125 + 546875 i - 1640625 + 21875 {(5 i)}^{3} + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.13

Apply the product rule to $5 i$ .

$78125 + 546875 i - 1640625 + 21875 (5^{3} i^{3}) + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.14

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

$78125 + 546875 i - 1640625 + 21875 (125 i^{3}) + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.15

Factor out $i^{2}$ .

$78125 + 546875 i - 1640625 + 21875 (125 (i^{2} \cdot i)) + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.16

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

$78125 + 546875 i - 1640625 + 21875 (125 (- 1 \cdot i)) + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.17

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

$78125 + 546875 i - 1640625 + 21875 (125 (- i)) + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.18

Multiply $- 1$ by $125$ .

$78125 + 546875 i - 1640625 + 21875 (- 125 i) + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.19

Multiply $- 125$ by $21875$ .

$78125 + 546875 i - 1640625 - 2734375 i + 35 \cdot 5^{3} {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.20

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

$78125 + 546875 i - 1640625 - 2734375 i + 35 \cdot 125 {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.21

Multiply $35$ by $125$ .

$78125 + 546875 i - 1640625 - 2734375 i + 4375 {(5 i)}^{4} + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.22

Apply the product rule to $5 i$ .

$78125 + 546875 i - 1640625 - 2734375 i + 4375 (5^{4} i^{4}) + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.23

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

$78125 + 546875 i - 1640625 - 2734375 i + 4375 (625 i^{4}) + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.24

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

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Step 2.1.24.1

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

$78125 + 546875 i - 1640625 - 2734375 i + 4375 (625 {(i^{2})}^{2}) + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.24.2

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

$78125 + 546875 i - 1640625 - 2734375 i + 4375 (625 {(- 1)}^{2}) + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.24.3

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

$78125 + 546875 i - 1640625 - 2734375 i + 4375 (625 \cdot 1) + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.25

Multiply $4375 (625 \cdot 1)$ .

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Step 2.1.25.1

Multiply $625$ by $1$ .

$78125 + 546875 i - 1640625 - 2734375 i + 4375 \cdot 625 + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.25.2

Multiply $4375$ by $625$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 21 \cdot 5^{2} {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.26

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 21 \cdot 25 {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.27

Multiply $21$ by $25$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 525 {(5 i)}^{5} + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.28

Apply the product rule to $5 i$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 525 (5^{5} i^{5}) + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.29

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 525 (3125 i^{5}) + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.30

Factor out $i^{4}$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 525 (3125 (i^{4} i)) + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.31

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

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Step 2.1.31.1

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 525 (3125 ({(i^{2})}^{2} i)) + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.31.2

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 525 (3125 ({(- 1)}^{2} i)) + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.31.3

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 525 (3125 (1 i)) + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.32

Multiply $i$ by $1$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 525 (3125 i) + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.33

Multiply $3125$ by $525$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 7 \cdot 5 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.34

Multiply $7$ by $5$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 {(5 i)}^{6} + {(5 i)}^{7}$

Step 2.1.35

Apply the product rule to $5 i$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 (5^{6} i^{6}) + {(5 i)}^{7}$

Step 2.1.36

Raise $5$ to the power of $6$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 (15625 i^{6}) + {(5 i)}^{7}$

Step 2.1.37

Factor out $i^{4}$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 (15625 (i^{4} i^{2})) + {(5 i)}^{7}$

Step 2.1.38

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

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Step 2.1.38.1

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 (15625 ({(i^{2})}^{2} i^{2})) + {(5 i)}^{7}$

Step 2.1.38.2

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 (15625 ({(- 1)}^{2} i^{2})) + {(5 i)}^{7}$

Step 2.1.38.3

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 (15625 (1 i^{2})) + {(5 i)}^{7}$

Step 2.1.39

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 (15625 i^{2}) + {(5 i)}^{7}$

Step 2.1.40

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 (15625 \cdot - 1) + {(5 i)}^{7}$

Step 2.1.41

Multiply $35 (15625 \cdot - 1)$ .

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Step 2.1.41.1

Multiply $15625$ by $- 1$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i + 35 \cdot - 15625 + {(5 i)}^{7}$

Step 2.1.41.2

Multiply $35$ by $- 15625$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + {(5 i)}^{7}$

Step 2.1.42

Apply the product rule to $5 i$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 5^{7} i^{7}$

Step 2.1.43

Raise $5$ to the power of $7$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 i^{7}$

Step 2.1.44

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

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Step 2.1.44.1

Factor out $i^{4}$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 (i^{4} i^{3})$

Step 2.1.44.2

Factor out $i^{2}$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 (i^{4} (i^{2} \cdot i))$

Step 2.1.45

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

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Step 2.1.45.1

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 ({(i^{2})}^{2} (i^{2} \cdot i))$

Step 2.1.45.2

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 ({(- 1)}^{2} (i^{2} \cdot i))$

Step 2.1.45.3

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 (1 (i^{2} \cdot i))$

Step 2.1.46

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 (i^{2} \cdot i)$

Step 2.1.47

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 (- 1 \cdot i)$

Step 2.1.48

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

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 + 78125 (- i)$

Step 2.1.49

Multiply $- 1$ by $78125$ .

$78125 + 546875 i - 1640625 - 2734375 i + 2734375 + 1640625 i - 546875 - 78125 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $1640625$ from $78125$ .

$- 1562500 + 546875 i - 2734375 i + 2734375 + 1640625 i - 546875 - 78125 i$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 1562500$ and $2734375$ .

$1171875 + 546875 i - 2734375 i + 1640625 i - 546875 - 78125 i$

Step 2.2.2.2

Subtract $546875$ from $1171875$ .

$625000 + 546875 i - 2734375 i + 1640625 i - 78125 i$

Step 2.2.3

Subtract $2734375 i$ from $546875 i$ .

$625000 - 2187500 i + 1640625 i - 78125 i$

Step 2.2.4

Add $- 2187500 i$ and $1640625 i$ .

$625000 - 546875 i - 78125 i$

Step 2.2.5

Subtract $78125 i$ from $- 546875 i$ .

$625000 - 625000 i$