Algebra Examples

${(\sqrt{2} + i \sqrt{2})}^{3}$

Step 1

Use the Binomial Theorem.

${\sqrt{2}}^{3} + 3 {\sqrt{2}}^{2} (i \sqrt{2}) + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$\sqrt{2^{3}} + 3 {\sqrt{2}}^{2} (i \sqrt{2}) + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.2

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

$\sqrt{8} + 3 {\sqrt{2}}^{2} (i \sqrt{2}) + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.3.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)} + 3 {\sqrt{2}}^{2} (i \sqrt{2}) + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.3.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 2} + 3 {\sqrt{2}}^{2} (i \sqrt{2}) + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.4

Pull terms out from under the radical.

$2 \sqrt{2} + 3 {\sqrt{2}}^{2} (i \sqrt{2}) + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.5

Multiply ${\sqrt{2}}^{2}$ by $\sqrt{2}$ by adding the exponents.

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Step 2.1.5.1

Move $\sqrt{2}$ .

$2 \sqrt{2} + 3 (\sqrt{2} {\sqrt{2}}^{2}) i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.5.2

Multiply $\sqrt{2}$ by ${\sqrt{2}}^{2}$ .

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Step 2.1.5.2.1

Raise $\sqrt{2}$ to the power of $1$ .

$2 \sqrt{2} + 3 ({\sqrt{2}}^{1} {\sqrt{2}}^{2}) i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.5.2.2

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

$2 \sqrt{2} + 3 {\sqrt{2}}^{1 + 2} i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.5.3

Add $1$ and $2$ .

$2 \sqrt{2} + 3 {\sqrt{2}}^{3} i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.6

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$2 \sqrt{2} + 3 \sqrt{2^{3}} i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.7

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

$2 \sqrt{2} + 3 \sqrt{8} i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.8

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.8.1

Factor $4$ out of $8$ .

$2 \sqrt{2} + 3 \sqrt{4 (2)} i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.8.2

Rewrite $4$ as $2^{2}$ .

$2 \sqrt{2} + 3 \sqrt{2^{2} \cdot 2} i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.9

Pull terms out from under the radical.

$2 \sqrt{2} + 3 (2 \sqrt{2}) i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.10

Multiply $2$ by $3$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 3 \sqrt{2} {(i \sqrt{2})}^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.11

Apply the product rule to $i \sqrt{2}$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 3 \sqrt{2} (i^{2} {\sqrt{2}}^{2}) + {(i \sqrt{2})}^{3}$

Step 2.1.12

Multiply $\sqrt{2}$ by ${\sqrt{2}}^{2}$ by adding the exponents.

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Step 2.1.12.1

Move ${\sqrt{2}}^{2}$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 3 ({\sqrt{2}}^{2} \sqrt{2}) i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.12.2

Multiply ${\sqrt{2}}^{2}$ by $\sqrt{2}$ .

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Step 2.1.12.2.1

Raise $\sqrt{2}$ to the power of $1$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 3 ({\sqrt{2}}^{2} {\sqrt{2}}^{1}) i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.12.2.2

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

$2 \sqrt{2} + 6 \sqrt{2} i + 3 {\sqrt{2}}^{2 + 1} i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.12.3

Add $2$ and $1$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 3 {\sqrt{2}}^{3} i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.13

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 3 \sqrt{2^{3}} i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.14

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

$2 \sqrt{2} + 6 \sqrt{2} i + 3 \sqrt{8} i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.15

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.15.1

Factor $4$ out of $8$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 3 \sqrt{4 (2)} i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.15.2

Rewrite $4$ as $2^{2}$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 3 \sqrt{2^{2} \cdot 2} i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.16

Pull terms out from under the radical.

$2 \sqrt{2} + 6 \sqrt{2} i + 3 (2 \sqrt{2}) i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.17

Multiply $2$ by $3$ .

$2 \sqrt{2} + 6 \sqrt{2} i + 6 \sqrt{2} i^{2} + {(i \sqrt{2})}^{3}$

Step 2.1.18

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

$2 \sqrt{2} + 6 \sqrt{2} i + 6 \sqrt{2} \cdot - 1 + {(i \sqrt{2})}^{3}$

Step 2.1.19

Multiply $- 1$ by $6$ .

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} + {(i \sqrt{2})}^{3}$

Step 2.1.20

Apply the product rule to $i \sqrt{2}$ .

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} + i^{3} {\sqrt{2}}^{3}$

Step 2.1.21

Factor out $i^{2}$ .

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} + i^{2} \cdot i {\sqrt{2}}^{3}$

Step 2.1.22

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

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} - 1 \cdot i {\sqrt{2}}^{3}$

Step 2.1.23

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

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} - i {\sqrt{2}}^{3}$

Step 2.1.24

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} - i \sqrt{2^{3}}$

Step 2.1.25

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

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} - i \sqrt{8}$

Step 2.1.26

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.26.1

Factor $4$ out of $8$ .

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} - i \sqrt{4 (2)}$

Step 2.1.26.2

Rewrite $4$ as $2^{2}$ .

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} - i \sqrt{2^{2} \cdot 2}$

Step 2.1.27

Pull terms out from under the radical.

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} - i (2 \sqrt{2})$

Step 2.1.28

Multiply $2$ by $- 1$ .

$2 \sqrt{2} + 6 \sqrt{2} i - 6 \sqrt{2} - 2 i \sqrt{2}$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $6 \sqrt{2}$ from $2 \sqrt{2}$ .

$6 \sqrt{2} i - 4 \sqrt{2} - 2 i \sqrt{2}$

Step 2.2.2

Reorder the factors of $6 \sqrt{2} i$ .

$6 i \sqrt{2} - 4 \sqrt{2} - 2 i \sqrt{2}$

Step 2.2.3

Subtract $2 i \sqrt{2}$ from $6 i \sqrt{2}$ .

$4 i \sqrt{2} - 4 \sqrt{2}$

Step 2.2.4

Reorder $4 i \sqrt{2}$ and $- 4 \sqrt{2}$ .

$- 4 \sqrt{2} + 4 i \sqrt{2}$

Step 3

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 4

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 5

Substitute the actual values of $a = - 4 \sqrt{2}$ and $b = 4 \sqrt{2}$ .

$| z | = \sqrt{{(4 \sqrt{2})}^{2} + {(- 4 \sqrt{2})}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Simplify the expression.

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Step 6.1.1

Apply the product rule to $4 \sqrt{2}$ .

$| z | = \sqrt{4^{2} {\sqrt{2}}^{2} + {(- 4 \sqrt{2})}^{2}}$

Step 6.1.2

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

$| z | = \sqrt{16 {\sqrt{2}}^{2} + {(- 4 \sqrt{2})}^{2}}$

Step 6.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$| z | = \sqrt{16 {(2^{\frac{1}{2}})}^{2} + {(- 4 \sqrt{2})}^{2}}$

Step 6.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \sqrt{16 \cdot 2^{\frac{1}{2} \cdot 2} + {(- 4 \sqrt{2})}^{2}}$

Step 6.2.3

Combine $\frac{1}{2}$ and $2$ .

$| z | = \sqrt{16 \cdot 2^{\frac{2}{2}} + {(- 4 \sqrt{2})}^{2}}$

Step 6.2.4

Cancel the common factor of $2$ .

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Step 6.2.4.1

Cancel the common factor.

$| z | = \sqrt{16 \cdot 2^{\frac{2}{2}} + {(- 4 \sqrt{2})}^{2}}$

Step 6.2.4.2

Rewrite the expression.

$| z | = \sqrt{16 \cdot 2 + {(- 4 \sqrt{2})}^{2}}$

Step 6.2.5

Evaluate the exponent.

$| z | = \sqrt{16 \cdot 2 + {(- 4 \sqrt{2})}^{2}}$

Step 6.3

Simplify the expression.

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Step 6.3.1

Multiply $16$ by $2$ .

$| z | = \sqrt{32 + {(- 4 \sqrt{2})}^{2}}$

Step 6.3.2

Apply the product rule to $- 4 \sqrt{2}$ .

$| z | = \sqrt{32 + {(- 4)}^{2} {\sqrt{2}}^{2}}$

Step 6.3.3

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

$| z | = \sqrt{32 + 16 {\sqrt{2}}^{2}}$

Step 6.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$| z | = \sqrt{32 + 16 {(2^{\frac{1}{2}})}^{2}}$

Step 6.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \sqrt{32 + 16 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 6.4.3

Combine $\frac{1}{2}$ and $2$ .

$| z | = \sqrt{32 + 16 \cdot 2^{\frac{2}{2}}}$

Step 6.4.4

Cancel the common factor of $2$ .

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Step 6.4.4.1

Cancel the common factor.

$| z | = \sqrt{32 + 16 \cdot 2^{\frac{2}{2}}}$

Step 6.4.4.2

Rewrite the expression.

$| z | = \sqrt{32 + 16 \cdot 2}$

Step 6.4.5

Evaluate the exponent.

$| z | = \sqrt{32 + 16 \cdot 2}$

Step 6.5

Simplify the expression.

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Step 6.5.1

Multiply $16$ by $2$ .

$| z | = \sqrt{32 + 32}$

Step 6.5.2

Add $32$ and $32$ .

$| z | = \sqrt{64}$

Step 6.5.3

Rewrite $64$ as $8^{2}$ .

$| z | = \sqrt{8^{2}}$

Step 6.5.4

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 8$

Step 7

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{4 \sqrt{2}}{- 4 \sqrt{2}})$

Step 8

Since inverse tangent of $\frac{4 \sqrt{2}}{- 4 \sqrt{2}}$ produces an angle in the second quadrant, the value of the angle is $\frac{3 π}{4}$ .

$θ = \frac{3 π}{4}$

Step 9

Substitute the values of $θ = \frac{3 π}{4}$ and $| z | = 8$ .

$8 (\cos (\frac{3 π}{4}) + i \sin (\frac{3 π}{4}))$