Algebra Examples

$\frac{5 (\cos (200) + i \sin (200))}{4 (\cos (50) + i \sin (50))}$

Step 1

Multiply the numerator and denominator of $\frac{5 (\cos (200) + i \sin (200))}{2.57115043 + 3.06417777 i}$ by the conjugate of $2.57115043 + 3.06417777 i$ to make the denominator real.

$\frac{5 (\cos (200) + i \sin (200))}{2.57115043 + 3.06417777 i} \cdot \frac{2.57115043 - 3.06417777 i}{2.57115043 - 3.06417777 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{5 (\cos (200) + i \sin (200)) (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Evaluate $\cos (200)$ .

$\frac{5 (- 0.93969262 + i \sin (200)) (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.1.2

Evaluate $\sin (200)$ .

$\frac{5 (- 0.93969262 + i \cdot - 0.34202014) (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.1.3

Move $- 0.34202014$ to the left of $i$ .

$\frac{5 (- 0.93969262 - 0.34202014 i) (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.2

Apply the distributive property.

$\frac{(5 \cdot - 0.93969262 + 5 (- 0.34202014 i)) (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.3

Multiply $5$ by $- 0.93969262$ .

$\frac{(- 4.6984631 + 5 (- 0.34202014 i)) (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.4

Multiply $- 0.34202014$ by $5$ .

$\frac{(- 4.6984631 - 1.71010071 i) (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.5

Expand $(- 4.6984631 - 1.71010071 i) (2.57115043 - 3.06417777 i)$ using the FOIL Method.

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Step 2.2.5.1

Apply the distributive property.

$\frac{- 4.6984631 (2.57115043 - 3.06417777 i) - 1.71010071 i (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.5.2

Apply the distributive property.

$\frac{- 4.6984631 \cdot 2.57115043 - 4.6984631 (- 3.06417777 i) - 1.71010071 i (2.57115043 - 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.5.3

Apply the distributive property.

$\frac{- 4.6984631 \cdot 2.57115043 - 4.6984631 (- 3.06417777 i) - 1.71010071 i \cdot 2.57115043 - 1.71010071 i (- 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6

Simplify and combine like terms.

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Step 2.2.6.1

Simplify each term.

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Step 2.2.6.1.1

Multiply $- 4.6984631$ by $2.57115043$ .

$\frac{- 12.08045547 - 4.6984631 (- 3.06417777 i) - 1.71010071 i \cdot 2.57115043 - 1.71010071 i (- 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.2

Multiply $- 3.06417777$ by $- 4.6984631$ .

$\frac{- 12.08045547 + 14.3969262 i - 1.71010071 i \cdot 2.57115043 - 1.71010071 i (- 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.3

Multiply $2.57115043$ by $- 1.71010071$ .

$\frac{- 12.08045547 + 14.3969262 i - 4.3969262 i - 1.71010071 i (- 3.06417777 i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.4

Multiply $- 1.71010071 i (- 3.06417777 i)$ .

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Step 2.2.6.1.4.1

Multiply $- 3.06417777$ by $- 1.71010071$ .

$\frac{- 12.08045547 + 14.3969262 i - 4.3969262 i + 5.2400526 i i}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.4.2

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

$\frac{- 12.08045547 + 14.3969262 i - 4.3969262 i + 5.2400526 (i^{1} i)}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.4.3

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

$\frac{- 12.08045547 + 14.3969262 i - 4.3969262 i + 5.2400526 (i^{1} i^{1})}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.4.4

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

$\frac{- 12.08045547 + 14.3969262 i - 4.3969262 i + 5.2400526 i^{1 + 1}}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.4.5

Add $1$ and $1$ .

$\frac{- 12.08045547 + 14.3969262 i - 4.3969262 i + 5.2400526 i^{2}}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.5

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

$\frac{- 12.08045547 + 14.3969262 i - 4.3969262 i + 5.2400526 \cdot - 1}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.1.6

Multiply $5.2400526$ by $- 1$ .

$\frac{- 12.08045547 + 14.3969262 i - 4.3969262 i - 5.2400526}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.2

Subtract $5.2400526$ from $- 12.08045547$ .

$\frac{- 17.32050807 + 14.3969262 i - 4.3969262 i}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.2.6.3

Subtract $4.3969262 i$ from $14.3969262 i$ .

$\frac{- 17.32050807 + 10 i}{(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(2.57115043 + 3.06417777 i) (2.57115043 - 3.06417777 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 17.32050807 + 10 i}{2.57115043 (2.57115043 - 3.06417777 i) + 3.06417777 i (2.57115043 - 3.06417777 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 17.32050807 + 10 i}{2.57115043 \cdot 2.57115043 + 2.57115043 (- 3.06417777 i) + 3.06417777 i (2.57115043 - 3.06417777 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 17.32050807 + 10 i}{2.57115043 \cdot 2.57115043 + 2.57115043 (- 3.06417777 i) + 3.06417777 i \cdot 2.57115043 + 3.06417777 i (- 3.06417777 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $2.57115043$ by $2.57115043$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 + 2.57115043 (- 3.06417777 i) + 3.06417777 i \cdot 2.57115043 + 3.06417777 i (- 3.06417777 i)}$

Step 2.3.2.2

Multiply $- 3.06417777$ by $2.57115043$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 - 7.87846202 i + 3.06417777 i \cdot 2.57115043 + 3.06417777 i (- 3.06417777 i)}$

Step 2.3.2.3

Multiply $2.57115043$ by $3.06417777$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 - 7.87846202 i + 7.87846202 i + 3.06417777 i (- 3.06417777 i)}$

Step 2.3.2.4

Multiply $- 3.06417777$ by $3.06417777$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 - 7.87846202 i + 7.87846202 i - 9.38918542 i i}$

Step 2.3.2.5

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

$\frac{- 17.32050807 + 10 i}{6.61081457 - 7.87846202 i + 7.87846202 i - 9.38918542 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 17.32050807 + 10 i}{6.61081457 - 7.87846202 i + 7.87846202 i - 9.38918542 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 17.32050807 + 10 i}{6.61081457 - 7.87846202 i + 7.87846202 i - 9.38918542 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 - 7.87846202 i + 7.87846202 i - 9.38918542 i^{2}}$

Step 2.3.2.9

Add $- 7.87846202 i$ and $7.87846202 i$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 + 0 i - 9.38918542 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Multiply $0$ by $i$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 + 0 - 9.38918542 i^{2}}$

Step 2.3.3.2

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

$\frac{- 17.32050807 + 10 i}{6.61081457 + 0 - 9.38918542 \cdot - 1}$

Step 2.3.3.3

Multiply $- 9.38918542$ by $- 1$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 + 0 + 9.38918542}$

Step 2.3.4

Add $6.61081457$ and $0$ .

$\frac{- 17.32050807 + 10 i}{6.61081457 + 9.38918542}$

Step 2.3.5

Add $6.61081457$ and $9.38918542$ .

$\frac{- 17.32050807 + 10 i}{16}$

Step 3

Rewrite $- 17.32050807$ as $1 (- 17.32050807)$ .

$\frac{1 (- 17.32050807) + 10 i}{16}$

Step 4

Factor $1$ out of $10 i$ .

$\frac{1 (- 17.32050807) + 1 (10 i)}{16}$

Step 5

Factor $1$ out of $1 (- 17.32050807) + 1 (10 i)$ .

$\frac{1 (- 17.32050807 + 10 i)}{16}$

Step 6

Factor $16$ out of $16$ .

$\frac{1 (- 17.32050807 + 10 i)}{16 (1)}$

Step 7

Separate fractions.

$\frac{1}{16} \cdot \frac{- 17.32050807 + 10 i}{1}$

Step 8

Simplify the expression.

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Step 8.1

Divide $1$ by $16$ .

$0.0625 \frac{- 17.32050807 + 10 i}{1}$

Step 8.2

Divide $- 17.32050807 + 10 i$ by $1$ .

$0.0625 (- 17.32050807 + 10 i)$

Step 9

Apply the distributive property.

$0.0625 \cdot - 17.32050807 + 0.0625 (10 i)$

Step 10

Multiply.

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Step 10.1

Multiply $0.0625$ by $- 17.32050807$ .

$- 1.08253175 + 0.0625 (10 i)$

Step 10.2

Multiply $10$ by $0.0625$ .

$- 1.08253175 + 0.625 i$