Algebra Examples

${(1 + \sqrt{3 i})}^{4}$

Step 1

Use the Binomial Theorem.

$1^{4} + 4 \cdot 1^{3} \sqrt{3 i} + 6 \cdot 1^{2} {\sqrt{3 i}}^{2} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

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.

$1 + 4 \cdot 1^{3} \sqrt{3 i} + 6 \cdot 1^{2} {\sqrt{3 i}}^{2} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.2

One to any power is one.

$1 + 4 \cdot 1 \sqrt{3 i} + 6 \cdot 1^{2} {\sqrt{3 i}}^{2} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.3

Multiply $4$ by $1$ .

$1 + 4 \sqrt{3 i} + 6 \cdot 1^{2} {\sqrt{3 i}}^{2} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.4

One to any power is one.

$1 + 4 \sqrt{3 i} + 6 \cdot 1 {\sqrt{3 i}}^{2} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.5

Multiply $6$ by $1$ .

$1 + 4 \sqrt{3 i} + 6 {\sqrt{3 i}}^{2} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.6

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

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Step 2.1.6.1

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

$1 + 4 \sqrt{3 i} + 6 {({(3 i)}^{\frac{1}{2}})}^{2} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.6.2

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

$1 + 4 \sqrt{3 i} + 6 {(3 i)}^{\frac{1}{2} \cdot 2} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.6.3

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

$1 + 4 \sqrt{3 i} + 6 {(3 i)}^{\frac{2}{2}} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

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.

$1 + 4 \sqrt{3 i} + 6 {(3 i)}^{\frac{2}{2}} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.6.4.2

Rewrite the expression.

$1 + 4 \sqrt{3 i} + 6 {(3 i)}^{1} + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.6.5

Simplify.

$1 + 4 \sqrt{3 i} + 6 (3 i) + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.7

Multiply $3$ by $6$ .

$1 + 4 \sqrt{3 i} + 18 i + 4 \cdot 1 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.8

Multiply $4$ by $1$ .

$1 + 4 \sqrt{3 i} + 18 i + 4 {\sqrt{3 i}}^{3} + {\sqrt{3 i}}^{4}$

Step 2.1.9

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

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{{(3 i)}^{3}} + {\sqrt{3 i}}^{4}$

Step 2.1.10

Apply the product rule to $3 i$ .

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{3^{3} i^{3}} + {\sqrt{3 i}}^{4}$

Step 2.1.11

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

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{27 i^{3}} + {\sqrt{3 i}}^{4}$

Step 2.1.12

Factor out $i^{2}$ .

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{27 (i^{2} \cdot i)} + {\sqrt{3 i}}^{4}$

Step 2.1.13

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

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{27 (- 1 \cdot i)} + {\sqrt{3 i}}^{4}$

Step 2.1.14

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

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{27 \cdot - 1 i} + {\sqrt{3 i}}^{4}$

Step 2.1.15

Combine exponents.

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Step 2.1.15.1

Factor out negative.

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{- (27 i)} + {\sqrt{3 i}}^{4}$

Step 2.1.15.2

Multiply $27$ by $- 1$ .

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{- 27 i} + {\sqrt{3 i}}^{4}$

Step 2.1.16

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

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Step 2.1.16.1

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

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{- 9 (3) i} + {\sqrt{3 i}}^{4}$

Step 2.1.16.2

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

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{{(3 i)}^{2} \cdot 3 i} + {\sqrt{3 i}}^{4}$

Step 2.1.16.3

Add parentheses.

$1 + 4 \sqrt{3 i} + 18 i + 4 \sqrt{{(3 i)}^{2} \cdot (3 i)} + {\sqrt{3 i}}^{4}$

Step 2.1.17

Pull terms out from under the radical.

$1 + 4 \sqrt{3 i} + 18 i + 4 (3 i \sqrt{3 i}) + {\sqrt{3 i}}^{4}$

Step 2.1.18

Multiply $3$ by $4$ .

$1 + 4 \sqrt{3 i} + 18 i + 12 (i \sqrt{3 i}) + {\sqrt{3 i}}^{4}$

Step 2.1.19

Rewrite ${\sqrt{3 i}}^{4}$ as ${(3 i)}^{2}$ .

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Step 2.1.19.1

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

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + {({(3 i)}^{\frac{1}{2}})}^{4}$

Step 2.1.19.2

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

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + {(3 i)}^{\frac{1}{2} \cdot 4}$

Step 2.1.19.3

Combine $\frac{1}{2}$ and $4$ .

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + {(3 i)}^{\frac{4}{2}}$

Step 2.1.19.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.19.4.1

Factor $2$ out of $4$ .

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + {(3 i)}^{\frac{2 \cdot 2}{2}}$

Step 2.1.19.4.2

Cancel the common factors.

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Step 2.1.19.4.2.1

Factor $2$ out of $2$ .

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + {(3 i)}^{\frac{2 \cdot 2}{2 (1)}}$

Step 2.1.19.4.2.2

Cancel the common factor.

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + {(3 i)}^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 2.1.19.4.2.3

Rewrite the expression.

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + {(3 i)}^{\frac{2}{1}}$

Step 2.1.19.4.2.4

Divide $2$ by $1$ .

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + {(3 i)}^{2}$

Step 2.1.20

Apply the product rule to $3 i$ .

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + 3^{2} i^{2}$

Step 2.1.21

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

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + 9 i^{2}$

Step 2.1.22

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

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} + 9 \cdot - 1$

Step 2.1.23

Multiply $9$ by $- 1$ .

$1 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i} - 9$

Step 2.2

Subtract $9$ from $1$ .

$- 8 + 4 \sqrt{3 i} + 18 i + 12 i \sqrt{3 i}$