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 each term.

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Step 2.1

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 2.2

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 2.3

Multiply $3$ by $1$ .

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

Step 2.4

Multiply $3$ by $- 1$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

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

Step 2.6

Multiply $3$ by $- 3$ .

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

Step 2.7

Rewrite ${\sqrt{3 i}}^{3}$ as $\sqrt{{(3 i)}^{3}}$ .

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

Step 2.8

Apply the product rule to $3 i$ .

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

Step 2.9

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

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

Step 2.10

Factor out $i^{2}$ .

$- 1 + 3 \sqrt{3 i} - 9 i + \sqrt{27 (i^{2} \cdot i)}$

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

Combine exponents.

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Step 2.13.1

Factor out negative.

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

Step 2.13.2

Multiply $27$ by $- 1$ .

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

Step 2.14

Rewrite $- 27 i$ as ${(3 i)}^{2} \cdot (3 i)$ .

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Step 2.14.1

Factor $- 9$ out of $- 27$ .

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

Step 2.14.2

Rewrite $- 9$ as ${(3 i)}^{2}$ .

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

Step 2.14.3

Add parentheses.

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

Step 2.15

Pull terms out from under the radical.

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