Pre-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 each term.

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Step 2.1

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 2.2

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 2.3

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 2.4

Multiply $- 1$ by $4$ .

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

Step 2.5

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 2.6

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 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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Step 2.10.1

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

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

Step 2.10.2

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

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

Step 2.10.3

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

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

Step 2.10.4

Cancel the common factor of $2$ .

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Step 2.10.4.1

Cancel the common factor.

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

Step 2.10.4.2

Rewrite the expression.

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

Step 2.10.5

Evaluate the exponent.

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

Step 2.11

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

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

Step 2.12

Multiply $6 (3 \cdot - 1)$ .

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Step 2.12.1

Multiply $3$ by $- 1$ .

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

Step 2.12.2

Multiply $6$ by $- 3$ .

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

Step 2.13

Multiply $4$ by $1$ .

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

Step 2.14

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

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Step 2.14.1

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

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

Step 2.14.2

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

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

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

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

Step 2.18

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

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Step 2.18.1

Factor $9$ out of $27$ .

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

Step 2.18.2

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

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

Step 2.19

Pull terms out from under the radical.

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

Step 2.20

Multiply $3$ by $- 1$ .

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

Step 2.21

Factor out $i^{2}$ .

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

Step 2.22

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

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

Step 2.23

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

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

Step 2.24

Multiply $- 1$ by $- 3$ .

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

Step 2.25

Multiply $3$ by $4$ .

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

Step 2.26

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

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Step 2.26.1

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

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

Step 2.26.2

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

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

Step 2.27

Raise $- 1$ to the power of $4$ .

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

Step 2.28

Multiply ${\sqrt{3}}^{4}$ by $1$ .

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

Step 2.29

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

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Step 2.29.1

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

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

Step 2.29.2

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

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

Step 2.29.3

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

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

Step 2.29.4

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

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Step 2.29.4.1

Factor $2$ out of $4$ .

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

Step 2.29.4.2

Cancel the common factors.

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Step 2.29.4.2.1

Factor $2$ out of $2$ .

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

Step 2.29.4.2.2

Cancel the common factor.

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

Step 2.29.4.2.3

Rewrite the expression.

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

Step 2.29.4.2.4

Divide $2$ by $1$ .

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

Step 2.30

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

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

Step 2.31

Rewrite $i^{4}$ as $1$ .

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Step 2.31.1

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

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

Step 2.31.2

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

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

Step 2.31.3

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

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

Step 2.32

Multiply $9$ by $1$ .

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

Step 3

Subtract $18$ from $1$ .

$- 4 \sqrt{3} i - 17 + 12 \sqrt{3} i + 9$

Step 4

Add $- 4 \sqrt{3} i$ and $12 \sqrt{3} i$ .

$8 \sqrt{3} i - 17 + 9$

Step 5

Add $- 17$ and $9$ .

$8 \sqrt{3} i - 8$

Step 6

Reorder $8 \sqrt{3} i$ and $- 8$ .

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