Finite Math Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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Step 2.1.5.1

Factor $9$ out of $27$ .

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

Step 2.1.5.2

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

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

Step 2.1.6

Pull terms out from under the radical.

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

Step 2.1.7

Multiply $3$ by $- 1$ .

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

Step 2.1.8

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

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

Step 2.1.9

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

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

Step 2.1.10

Multiply ${\sqrt{3}}^{2}$ by $1$ .

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

Step 2.1.11

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

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Step 2.1.11.1

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

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

Step 2.1.11.2

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

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

Step 2.1.11.3

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

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

Step 2.1.11.4

Cancel the common factor of $2$ .

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Step 2.1.11.4.1

Cancel the common factor.

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

Step 2.1.11.4.2

Rewrite the expression.

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

Step 2.1.11.5

Evaluate the exponent.

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

Step 2.1.12

Multiply $3$ by $3$ .

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

Step 2.1.13

Multiply $- 1$ by $3$ .

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

Step 2.1.14

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

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

Step 2.1.15

Multiply $- 1$ by $- 3$ .

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

Step 2.1.16

Factor out $i^{2}$ .

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

Step 2.1.17

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

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

Step 2.1.18

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

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

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

$0 + 9 i - i$

Step 2.2.2

Add $0$ and $9 i$ .

$9 i - i$

Step 2.2.3

Subtract $i$ from $9 i$ .

$8 i$