Precalculus Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

Multiply $3$ by $1$ .

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

Step 2.1.4

Multiply $3$ by $- 1$ .

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

Step 2.1.5

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

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

Step 2.1.6

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

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

Rewrite the expression.

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

Step 2.1.6.5

Evaluate the exponent.

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

Step 2.1.7

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

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

Step 2.1.8

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

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Step 2.1.8.1

Multiply $3$ by $- 1$ .

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

Step 2.1.8.2

Multiply $- 3$ by $- 3$ .

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

Step 2.1.9

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

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

Step 2.1.10

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

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

Step 2.1.11

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

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

Step 2.1.12

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

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Step 2.1.12.1

Factor $9$ out of $27$ .

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

Step 2.1.12.2

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

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

Step 2.1.13

Pull terms out from under the radical.

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

Step 2.1.14

Factor out $i^{2}$ .

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

Step 2.1.15

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

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

Step 2.1.16

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

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

Step 2.1.17

Multiply $- 1$ by $3$ .

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 1$ and $9$ .

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

Step 2.2.2

Subtract $3 \sqrt{3} i$ from $3 \sqrt{3} i$ .

$0 + 8$

Step 2.2.3

Add $0$ and $8$ .

$8$