Trigonometry Examples

${(\sqrt{2} - i)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite

${\sqrt{2}}^{4}$ as

$2^{2}$ .

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Step 2.1.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

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

Step 2.1.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.1.3

Combine

$\frac{1}{2}$ and

$4$ .

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

Step 2.1.1.4

Cancel the common factor of

$4$ and

$2$ .

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Step 2.1.1.4.1

Factor

$2$ out of

$4$ .

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

Step 2.1.1.4.2

Cancel the common factors.

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Step 2.1.1.4.2.1

Factor

$2$ out of

$2$ .

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

Step 2.1.1.4.2.2

Cancel the common factor.

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

Step 2.1.1.4.2.3

Rewrite the expression.

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

Step 2.1.1.4.2.4

Divide

$2$ by

$1$ .

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

Step 2.1.2

Raise

$2$ to the power of

$2$ .

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

Step 2.1.3

Rewrite

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

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

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

Step 2.1.4

Raise

$2$ to the power of

$3$ .

$4 + 4 \sqrt{8} (- i) + 6 {\sqrt{2}}^{2} {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.5

Rewrite

$8$ as

$2^{2} \cdot 2$ .

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Step 2.1.5.1

Factor

$4$ out of

$8$ .

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

Step 2.1.5.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 2.1.6

Pull terms out from under the radical.

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

Step 2.1.7

Multiply

$2$ by

$4$ .

$4 + 8 \sqrt{2} (- i) + 6 {\sqrt{2}}^{2} {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.8

Multiply

$- 1$ by

$8$ .

$4 - 8 \sqrt{2} i + 6 {\sqrt{2}}^{2} {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 2.1.9.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$4 - 8 \sqrt{2} i + 6 {(2^{\frac{1}{2}})}^{2} {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$4 - 8 \sqrt{2} i + 6 \cdot 2^{\frac{1}{2} \cdot 2} {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.3

Combine

$\frac{1}{2}$ and

$2$ .

$4 - 8 \sqrt{2} i + 6 \cdot 2^{\frac{2}{2}} {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.4

Cancel the common factor of

$2$ .

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Step 2.1.9.4.1

Cancel the common factor.

$4 - 8 \sqrt{2} i + 6 \cdot 2^{\frac{2}{2}} {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.4.2

Rewrite the expression.

$4 - 8 \sqrt{2} i + 6 \cdot 2^{1} {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.5

Evaluate the exponent.

$4 - 8 \sqrt{2} i + 6 \cdot 2 {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.10

Multiply

$6$ by

$2$ .

$4 - 8 \sqrt{2} i + 12 {(- i)}^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.11

Apply the product rule to

$- i$ .

$4 - 8 \sqrt{2} i + 12 ({(- 1)}^{2} i^{2}) + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.12

Raise

$- 1$ to the power of

$2$ .

$4 - 8 \sqrt{2} i + 12 (1 i^{2}) + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.13

Multiply

$i^{2}$ by

$1$ .

$4 - 8 \sqrt{2} i + 12 i^{2} + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.14

Rewrite

$i^{2}$ as

$- 1$ .

$4 - 8 \sqrt{2} i + 12 \cdot - 1 + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.15

Multiply

$12$ by

$- 1$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.16

Apply the product rule to

$- i$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} ({(- 1)}^{3} i^{3}) + {(- i)}^{4}$

Step 2.1.17

Raise

$- 1$ to the power of

$3$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} (- i^{3}) + {(- i)}^{4}$

Step 2.1.18

Factor out

$i^{2}$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} (- (i^{2} \cdot i)) + {(- i)}^{4}$

Step 2.1.19

Rewrite

$i^{2}$ as

$- 1$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} (- (- 1 \cdot i)) + {(- i)}^{4}$

Step 2.1.20

Rewrite

$- 1 i$ as

$- i$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} (- - i) + {(- i)}^{4}$

Step 2.1.21

Multiply

$- 1$ by

$- 1$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} (1 i) + {(- i)}^{4}$

Step 2.1.22

Multiply

$i$ by

$1$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} i + {(- i)}^{4}$

Step 2.1.23

Apply the product rule to

$- i$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} i + {(- 1)}^{4} i^{4}$

Step 2.1.24

Raise

$- 1$ to the power of

$4$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} i + 1 i^{4}$

Step 2.1.25

Multiply

$i^{4}$ by

$1$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} i + i^{4}$

Step 2.1.26

Rewrite

$i^{4}$ as

$1$ .

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Step 2.1.26.1

Rewrite

$i^{4}$ as

${(i^{2})}^{2}$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} i + {(i^{2})}^{2}$

Step 2.1.26.2

Rewrite

$i^{2}$ as

$- 1$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} i + {(- 1)}^{2}$

Step 2.1.26.3

Raise

$- 1$ to the power of

$2$ .

$4 - 8 \sqrt{2} i - 12 + 4 \sqrt{2} i + 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract

$12$ from

$4$ .

$- 8 \sqrt{2} i - 8 + 4 \sqrt{2} i + 1$

Step 2.2.2

Add

$- 8 \sqrt{2} i$ and

$4 \sqrt{2} i$ .

$- 4 \sqrt{2} i - 8 + 1$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Add

$- 8$ and

$1$ .

$- 4 \sqrt{2} i - 7$

Step 2.2.3.2

Reorder

$- 4 \sqrt{2} i$ and

$- 7$ .

$- 7 - 4 \sqrt{2} i$