Algebra Examples

$6 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2})) \cdot 4 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$6 (0 + i \sin (\frac{π}{2})) \cdot 4 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$6 (0 + i \cdot 1) \cdot 4 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.3

Multiply $i$ by $1$ .

$6 (0 + i) \cdot 4 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 2

Simplify the expression.

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Step 2.1

Add $0$ and $i$ .

$6 i \cdot 4 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 2.2

Multiply $4$ by $6$ .

$24 i (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3

Simplify each term.

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Step 3.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$24 i (\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4}))$

Step 3.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$24 i (\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})$

Step 3.3

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$24 i (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 4

Simplify terms.

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Step 4.1

Apply the distributive property.

$24 i \frac{\sqrt{2}}{2} + 24 i \frac{i \sqrt{2}}{2}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $24 i$ .

$2 (12 i) \frac{\sqrt{2}}{2} + 24 i \frac{i \sqrt{2}}{2}$

Step 4.2.2

Cancel the common factor.

$2 (12 i) \frac{\sqrt{2}}{2} + 24 i \frac{i \sqrt{2}}{2}$

Step 4.2.3

Rewrite the expression.

$12 i \sqrt{2} + 24 i \frac{i \sqrt{2}}{2}$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Factor $2$ out of $24 i$ .

$12 i \sqrt{2} + 2 (12 i) \frac{i \sqrt{2}}{2}$

Step 4.3.2

Cancel the common factor.

$12 i \sqrt{2} + 2 (12 i) \frac{i \sqrt{2}}{2}$

Step 4.3.3

Rewrite the expression.

$12 i \sqrt{2} + 12 i (i \sqrt{2})$

Step 5

Raise $i$ to the power of $1$ .

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

Step 6

Raise $i$ to the power of $1$ .

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

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 i \sqrt{2} + 12 i^{1 + 1} \sqrt{2}$

Step 8

Add $1$ and $1$ .

$12 i \sqrt{2} + 12 i^{2} \sqrt{2}$

Step 9

Simplify each term.

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Step 9.1

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

$12 i \sqrt{2} + 12 \cdot - 1 \sqrt{2}$

Step 9.2

Multiply $12$ by $- 1$ .

$12 i \sqrt{2} - 12 \sqrt{2}$

Step 10

Reorder $12 i \sqrt{2}$ and $- 12 \sqrt{2}$ .

$- 12 \sqrt{2} + 12 i \sqrt{2}$