Trigonometry Examples

${(\sqrt{10} + i)}^{4}$

Step 1

Use the Binomial Theorem.

${\sqrt{10}}^{4} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${\sqrt{10}}^{4}$ as $10^{2}$ .

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Step 2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

${(10^{\frac{1}{2}})}^{4} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.1.2

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

$10^{\frac{1}{2} \cdot 4} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.1.3

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

$10^{\frac{4}{2}} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

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 $4$ .

$10^{\frac{2 \cdot 2}{2}} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

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 $2$ .

$10^{\frac{2 \cdot 2}{2 (1)}} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.1.4.2.2

Cancel the common factor.

$10^{\frac{2 \cdot 2}{2 \cdot 1}} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.1.4.2.3

Rewrite the expression.

$10^{\frac{2}{1}} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.1.4.2.4

Divide $2$ by $1$ .

$10^{2} + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.2

Raise $10$ to the power of $2$ .

$100 + 4 {\sqrt{10}}^{3} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.3

Rewrite ${\sqrt{10}}^{3}$ as $\sqrt{10^{3}}$ .

$100 + 4 \sqrt{10^{3}} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.4

Raise $10$ to the power of $3$ .

$100 + 4 \sqrt{1000} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.5

Rewrite $1000$ as $10^{2} \cdot 10$ .

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Step 2.1.5.1

Factor $100$ out of $1000$ .

$100 + 4 \sqrt{100 (10)} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.5.2

Rewrite $100$ as $10^{2}$ .

$100 + 4 \sqrt{10^{2} \cdot 10} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.6

Pull terms out from under the radical.

$100 + 4 (10 \sqrt{10}) i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.7

Multiply $10$ by $4$ .

$100 + 40 \sqrt{10} i + 6 {\sqrt{10}}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.8

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$100 + 40 \sqrt{10} i + 6 {(10^{\frac{1}{2}})}^{2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.8.2

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

$100 + 40 \sqrt{10} i + 6 \cdot 10^{\frac{1}{2} \cdot 2} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.8.3

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

$100 + 40 \sqrt{10} i + 6 \cdot 10^{\frac{2}{2}} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.8.4

Cancel the common factor of $2$ .

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Step 2.1.8.4.1

Cancel the common factor.

$100 + 40 \sqrt{10} i + 6 \cdot 10^{\frac{2}{2}} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.8.4.2

Rewrite the expression.

$100 + 40 \sqrt{10} i + 6 \cdot 10^{1} i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.8.5

Evaluate the exponent.

$100 + 40 \sqrt{10} i + 6 \cdot 10 i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.9

Multiply $6$ by $10$ .

$100 + 40 \sqrt{10} i + 60 i^{2} + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.10

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

$100 + 40 \sqrt{10} i + 60 \cdot - 1 + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.11

Multiply $60$ by $- 1$ .

$100 + 40 \sqrt{10} i - 60 + 4 \sqrt{10} i^{3} + i^{4}$

Step 2.1.12

Factor out $i^{2}$ .

$100 + 40 \sqrt{10} i - 60 + 4 \sqrt{10} (i^{2} \cdot i) + i^{4}$

Step 2.1.13

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

$100 + 40 \sqrt{10} i - 60 + 4 \sqrt{10} (- 1 \cdot i) + i^{4}$

Step 2.1.14

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

$100 + 40 \sqrt{10} i - 60 + 4 \sqrt{10} (- i) + i^{4}$

Step 2.1.15

Multiply $- 1$ by $4$ .

$100 + 40 \sqrt{10} i - 60 - 4 \sqrt{10} i + i^{4}$

Step 2.1.16

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

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Step 2.1.16.1

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

$100 + 40 \sqrt{10} i - 60 - 4 \sqrt{10} i + {(i^{2})}^{2}$

Step 2.1.16.2

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

$100 + 40 \sqrt{10} i - 60 - 4 \sqrt{10} i + {(- 1)}^{2}$

Step 2.1.16.3

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

$100 + 40 \sqrt{10} i - 60 - 4 \sqrt{10} i + 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $60$ from $100$ .

$40 \sqrt{10} i + 40 - 4 \sqrt{10} i + 1$

Step 2.2.2

Subtract $4 \sqrt{10} i$ from $40 \sqrt{10} i$ .

$36 \sqrt{10} i + 40 + 1$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Add $40$ and $1$ .

$36 \sqrt{10} i + 41$

Step 2.2.3.2

Reorder $36 \sqrt{10} i$ and $41$ .

$41 + 36 \sqrt{10} i$