Algebra Examples

$6 (\cos (\frac{3 π}{4}) + i \sin (\frac{3 π}{4})) \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1

Simplify the numerator.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$6 (- \cos (\frac{π}{4}) + i \sin (\frac{3 π}{4})) \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.2

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

$6 (- \frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{4})) \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$6 (- \frac{\sqrt{2}}{2} + i \sin (\frac{π}{4})) \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.4

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

$6 (- \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}) \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.5

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

$6 (- \frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2}) \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.6

Combine the numerators over the common denominator.

$6 \frac{- \sqrt{2} + i \sqrt{2}}{2} \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 2

Combine $6$ and $\frac{- \sqrt{2} + i \sqrt{2}}{2}$ .

$\frac{6 (- \sqrt{2} + i \sqrt{2})}{2} \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Reduce the expression $\frac{6 (- \sqrt{2} + i \sqrt{2})}{2}$ by cancelling the common factors.

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Step 3.1.1

Factor $2$ out of $6 (- \sqrt{2} + i \sqrt{2})$ .

$\frac{2 (3 (- \sqrt{2} + i \sqrt{2}))}{2} \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3.1.2

Factor $2$ out of $2$ .

$\frac{2 (3 (- \sqrt{2} + i \sqrt{2}))}{2 (1)} \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3.1.3

Cancel the common factor.

$\frac{2 (3 (- \sqrt{2} + i \sqrt{2}))}{2 \cdot 1} \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3.1.4

Rewrite the expression.

$\frac{3 (- \sqrt{2} + i \sqrt{2})}{1} \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3.2

Divide $3 (- \sqrt{2} + i \sqrt{2})$ by $1$ .

$3 (- \sqrt{2} + i \sqrt{2}) \div 2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 4

Rewrite the division as a fraction.

$\frac{3 (- \sqrt{2} + i \sqrt{2})}{2} (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 5

Simplify each term.

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Step 5.1

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

$\frac{3 (- \sqrt{2} + i \sqrt{2})}{2} (\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4}))$

Step 5.2

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

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

Step 5.3

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

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

Step 6

Apply the distributive property.

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

Step 7

Multiply $\frac{3 (- \sqrt{2} + i \sqrt{2})}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 7.1

Multiply $\frac{3 (- \sqrt{2} + i \sqrt{2})}{2}$ by $\frac{\sqrt{2}}{2}$ .

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

Step 7.2

Multiply $2$ by $2$ .

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

Step 8

Multiply $\frac{3 (- \sqrt{2} + i \sqrt{2})}{2} \cdot \frac{i \sqrt{2}}{2}$ .

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Step 8.1

Multiply $\frac{3 (- \sqrt{2} + i \sqrt{2})}{2}$ by $\frac{i \sqrt{2}}{2}$ .

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

Step 8.2

Multiply $2$ by $2$ .

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify each term.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Multiply $- 1$ by $3$ .

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

Step 10.3

Apply the distributive property.

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

Step 10.4

Multiply $- 3 \sqrt{2} \sqrt{2}$ .

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Step 10.4.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.4.3

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

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

Step 10.4.4

Add $1$ and $1$ .

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

Step 10.5

Multiply $3 i \sqrt{2} \sqrt{2}$ .

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Step 10.5.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.5.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.5.3

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

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

Step 10.5.4

Add $1$ and $1$ .

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

Step 10.6

Simplify each term.

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Step 10.6.1

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

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Step 10.6.1.1

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

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

Step 10.6.1.2

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

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

Step 10.6.1.3

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

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

Step 10.6.1.4

Cancel the common factor of $2$ .

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Step 10.6.1.4.1

Cancel the common factor.

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

Step 10.6.1.4.2

Rewrite the expression.

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

Step 10.6.1.5

Evaluate the exponent.

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

Step 10.6.2

Multiply $- 3$ by $2$ .

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

Step 10.6.3

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

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Step 10.6.3.1

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

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

Step 10.6.3.2

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

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

Step 10.6.3.3

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

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

Step 10.6.3.4

Cancel the common factor of $2$ .

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Step 10.6.3.4.1

Cancel the common factor.

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

Step 10.6.3.4.2

Rewrite the expression.

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

Step 10.6.3.5

Evaluate the exponent.

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

Step 10.6.4

Multiply $2$ by $3$ .

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

Step 10.7

Apply the distributive property.

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

Step 10.8

Multiply $- 1$ by $3$ .

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

Step 10.9

Apply the distributive property.

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

Step 10.10

Multiply $- 3 \sqrt{2} (i \sqrt{2})$ .

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Step 10.10.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.10.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.10.3

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

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

Step 10.10.4

Add $1$ and $1$ .

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

Step 10.11

Multiply $3 i \sqrt{2} (i \sqrt{2})$ .

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Step 10.11.1

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

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

Step 10.11.2

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

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

Step 10.11.3

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

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

Step 10.11.4

Add $1$ and $1$ .

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

Step 10.11.5

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.11.6

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.11.7

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

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

Step 10.11.8

Add $1$ and $1$ .

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

Step 10.12

Simplify each term.

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Step 10.12.1

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

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Step 10.12.1.1

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

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

Step 10.12.1.2

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

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

Step 10.12.1.3

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

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

Step 10.12.1.4

Cancel the common factor of $2$ .

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Step 10.12.1.4.1

Cancel the common factor.

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

Step 10.12.1.4.2

Rewrite the expression.

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

Step 10.12.1.5

Evaluate the exponent.

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

Step 10.12.2

Multiply $- 3$ by $2$ .

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

Step 10.12.3

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

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

Step 10.12.4

Multiply $3$ by $- 1$ .

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

Step 10.12.5

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

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Step 10.12.5.1

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

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

Step 10.12.5.2

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

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

Step 10.12.5.3

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

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

Step 10.12.5.4

Cancel the common factor of $2$ .

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Step 10.12.5.4.1

Cancel the common factor.

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

Step 10.12.5.4.2

Rewrite the expression.

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

Step 10.12.5.5

Evaluate the exponent.

$\frac{- 6 + 6 i - 6 i - 3 \cdot 2}{4}$

Step 10.12.6

Multiply $- 3$ by $2$ .

$\frac{- 6 + 6 i - 6 i - 6}{4}$

Step 11

Simplify by adding terms.

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Step 11.1

Subtract $6$ from $- 6$ .

$\frac{- 12 + 6 i - 6 i}{4}$

Step 11.2

Subtract $6 i$ from $6 i$ .

$\frac{- 12 + 0}{4}$

Step 11.3

Simplify the expression.

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Step 11.3.1

Add $- 12$ and $0$ .

$\frac{- 12}{4}$

Step 11.3.2

Divide $- 12$ by $4$ .

$- 3$