Calculus Examples

${(1 - i \sqrt{3})}^{3}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

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

Step 2.1.2

One to any power is one.

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

Step 2.1.3

Multiply $3$ by $1$ .

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

Step 2.1.4

Multiply $- 1$ by $3$ .

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

Step 2.1.5

Multiply $3$ by $1$ .

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

Step 2.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.6.1

Apply the product rule to $- i \sqrt{3}$ .

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

Step 2.1.6.2

Apply the product rule to $- i$ .

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

Step 2.1.7

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

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

Step 2.1.8

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

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

Step 2.1.9

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

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

Step 2.1.10

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

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Step 2.1.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Cancel the common factor of $2$ .

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Step 2.1.10.4.1

Cancel the common factor.

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

Step 2.1.10.4.2

Rewrite the expression.

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

Step 2.1.10.5

Evaluate the exponent.

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

Step 2.1.11

Multiply $3 (- 1 \cdot 3)$ .

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Step 2.1.11.1

Multiply $- 1$ by $3$ .

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

Step 2.1.11.2

Multiply $3$ by $- 3$ .

$1 - 3 i \sqrt{3} - 9 + {(- i \sqrt{3})}^{3}$

Step 2.1.12

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.12.1

Apply the product rule to $- i \sqrt{3}$ .

$1 - 3 i \sqrt{3} - 9 + {(- i)}^{3} {\sqrt{3}}^{3}$

Step 2.1.12.2

Apply the product rule to $- i$ .

$1 - 3 i \sqrt{3} - 9 + {(- 1)}^{3} i^{3} {\sqrt{3}}^{3}$

Step 2.1.13

Raise $- 1$ to the power of $3$ .

$1 - 3 i \sqrt{3} - 9 - i^{3} {\sqrt{3}}^{3}$

Step 2.1.14

Factor out $i^{2}$ .

$1 - 3 i \sqrt{3} - 9 - (i^{2} \cdot i) {\sqrt{3}}^{3}$

Step 2.1.15

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

$1 - 3 i \sqrt{3} - 9 - (- 1 \cdot i) {\sqrt{3}}^{3}$

Step 2.1.16

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

$1 - 3 i \sqrt{3} - 9 - - i {\sqrt{3}}^{3}$

Step 2.1.17

Multiply $- 1$ by $- 1$ .

$1 - 3 i \sqrt{3} - 9 + 1 i {\sqrt{3}}^{3}$

Step 2.1.18

Multiply $i$ by $1$ .

$1 - 3 i \sqrt{3} - 9 + i {\sqrt{3}}^{3}$

Step 2.1.19

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

$1 - 3 i \sqrt{3} - 9 + i \sqrt{3^{3}}$

Step 2.1.20

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

$1 - 3 i \sqrt{3} - 9 + i \sqrt{27}$

Step 2.1.21

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.1.21.1

Factor $9$ out of $27$ .

$1 - 3 i \sqrt{3} - 9 + i \sqrt{9 (3)}$

Step 2.1.21.2

Rewrite $9$ as $3^{2}$ .

$1 - 3 i \sqrt{3} - 9 + i \sqrt{3^{2} \cdot 3}$

Step 2.1.22

Pull terms out from under the radical.

$1 - 3 i \sqrt{3} - 9 + i (3 \sqrt{3})$

Step 2.1.23

Move $3$ to the left of $i$ .

$1 - 3 i \sqrt{3} - 9 + 3 i \sqrt{3}$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $9$ from $1$ .

$- 3 i \sqrt{3} - 8 + 3 i \sqrt{3}$

Step 2.2.2

Add $- 3 i \sqrt{3}$ and $3 i \sqrt{3}$ .

$0 - 8$

Step 2.2.3

Subtract $8$ from $0$ .

$- 8$

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 $a = - 8$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(- 8)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(- 8)}^{2}}$

Step 6.2

Raise $- 8$ to the power of $2$ .

$| z | = \sqrt{0 + 64}$

Step 6.3

Add $0$ and $64$ .

$| z | = \sqrt{64}$

Step 6.4

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

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

Step 6.5

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.

$θ = \arctan (\frac{0}{- 8})$

Step 8

Since inverse tangent of $\frac{0}{- 8}$ produces an angle in the second quadrant, the value of the angle is $π$ .

$θ = π$

Step 9

Substitute the values of $θ = π$ and $| z | = 8$ .

$8 (\cos (π) + i \sin (π))$