Finite Math Examples

${(2 + 5 i)}^{2} + (3 + 4 i) + \frac{i}{\cos (30) + i \sin (30)}$

Step 1

Remove parentheses.

${(2 + 5 i)}^{2} + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2

Simplify each term.

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Step 2.1

Rewrite ${(2 + 5 i)}^{2}$ as $(2 + 5 i) (2 + 5 i)$ .

$(2 + 5 i) (2 + 5 i) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.2

Expand $(2 + 5 i) (2 + 5 i)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$2 (2 + 5 i) + 5 i (2 + 5 i) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.2.2

Apply the distributive property.

$2 \cdot 2 + 2 (5 i) + 5 i (2 + 5 i) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.2.3

Apply the distributive property.

$2 \cdot 2 + 2 (5 i) + 5 i \cdot 2 + 5 i (5 i) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $2$ by $2$ .

$4 + 2 (5 i) + 5 i \cdot 2 + 5 i (5 i) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.2

Multiply $5$ by $2$ .

$4 + 10 i + 5 i \cdot 2 + 5 i (5 i) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.3

Multiply $2$ by $5$ .

$4 + 10 i + 10 i + 5 i (5 i) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.4

Multiply $5 i (5 i)$ .

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Step 2.3.1.4.1

Multiply $5$ by $5$ .

$4 + 10 i + 10 i + 25 i i + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.4.2

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

$4 + 10 i + 10 i + 25 (i^{1} i) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.4.3

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

$4 + 10 i + 10 i + 25 (i^{1} i^{1}) + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.4.4

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

$4 + 10 i + 10 i + 25 i^{1 + 1} + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.4.5

Add $1$ and $1$ .

$4 + 10 i + 10 i + 25 i^{2} + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.5

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

$4 + 10 i + 10 i + 25 \cdot - 1 + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.1.6

Multiply $25$ by $- 1$ .

$4 + 10 i + 10 i - 25 + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.2

Subtract $25$ from $4$ .

$- 21 + 10 i + 10 i + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.3.3

Add $10 i$ and $10 i$ .

$- 21 + 20 i + 3 + 4 i + \frac{i}{\cos (30) + i \sin (30)}$

Step 2.4

Multiply the numerator and denominator of $\frac{i}{0.8660254 + 0.5 i}$ by the conjugate of $0.8660254 + 0.5 i$ to make the denominator real.

$- 21 + 20 i + 3 + 4 i + \frac{i}{0.8660254 + 0.5 i} \cdot \frac{0.8660254 - 0.5 i}{0.8660254 - 0.5 i}$

Step 2.5

Multiply.

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Step 2.5.1

Combine.

$- 21 + 20 i + 3 + 4 i + \frac{i (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2

Simplify the numerator.

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Step 2.5.2.1

Apply the distributive property.

$- 21 + 20 i + 3 + 4 i + \frac{i \cdot 0.8660254 + i (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2.2

Move $0.8660254$ to the left of $i$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.8660254 \cdot i + i (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2.3

Multiply $i (- 0.5 i)$ .

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Step 2.5.2.3.1

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

$- 21 + 20 i + 3 + 4 i + \frac{0.8660254 \cdot i - 0.5 (i^{1} i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2.3.2

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

$- 21 + 20 i + 3 + 4 i + \frac{0.8660254 \cdot i - 0.5 (i^{1} i^{1})}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2.3.3

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

$- 21 + 20 i + 3 + 4 i + \frac{0.8660254 \cdot i - 0.5 i^{1 + 1}}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2.3.4

Add $1$ and $1$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.8660254 \cdot i - 0.5 i^{2}}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2.4

Simplify each term.

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Step 2.5.2.4.1

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

$- 21 + 20 i + 3 + 4 i + \frac{0.8660254 i - 0.5 \cdot - 1}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2.4.2

Multiply $- 0.5$ by $- 1$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.8660254 i + 0.5}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.2.5

Reorder $0.8660254 i$ and $0.5$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 2.5.3

Simplify the denominator.

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Step 2.5.3.1

Expand $(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)$ using the FOIL Method.

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Step 2.5.3.1.1

Apply the distributive property.

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.8660254 (0.8660254 - 0.5 i) + 0.5 i (0.8660254 - 0.5 i)}$

Step 2.5.3.1.2

Apply the distributive property.

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.8660254 \cdot 0.8660254 + 0.8660254 (- 0.5 i) + 0.5 i (0.8660254 - 0.5 i)}$

Step 2.5.3.1.3

Apply the distributive property.

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.8660254 \cdot 0.8660254 + 0.8660254 (- 0.5 i) + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Step 2.5.3.2

Simplify.

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Step 2.5.3.2.1

Multiply $0.8660254$ by $0.8660254$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 + 0.8660254 (- 0.5 i) + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Step 2.5.3.2.2

Multiply $- 0.5$ by $0.8660254$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 - 0.4330127 i + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Step 2.5.3.2.3

Multiply $0.8660254$ by $0.5$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 - 0.4330127 i + 0.4330127 i + 0.5 i (- 0.5 i)}$

Step 2.5.3.2.4

Multiply $- 0.5$ by $0.5$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i i}$

Step 2.5.3.2.5

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

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 (i^{1} i)}$

Step 2.5.3.2.6

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

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 (i^{1} i^{1})}$

Step 2.5.3.2.7

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

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i^{1 + 1}}$

Step 2.5.3.2.8

Add $1$ and $1$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i^{2}}$

Step 2.5.3.2.9

Add $- 0.4330127 i$ and $0.4330127 i$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 + 0 i - 0.25 i^{2}}$

Step 2.5.3.3

Simplify each term.

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Step 2.5.3.3.1

Multiply $0$ by $i$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 + 0 - 0.25 i^{2}}$

Step 2.5.3.3.2

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

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 + 0 - 0.25 \cdot - 1}$

Step 2.5.3.3.3

Multiply $- 0.25$ by $- 1$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 + 0 + 0.25}$

Step 2.5.3.4

Add $0.75$ and $0$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{0.75 + 0.25}$

Step 2.5.3.5

Add $0.75$ and $0.25$ .

$- 21 + 20 i + 3 + 4 i + \frac{0.5 + 0.8660254 i}{1}$

Step 2.6

Rewrite $0.5$ as $1 (0.5)$ .

$- 21 + 20 i + 3 + 4 i + \frac{1 (0.5) + 0.8660254 i}{1}$

Step 2.7

Factor $1$ out of $0.8660254 i$ .

$- 21 + 20 i + 3 + 4 i + \frac{1 (0.5) + 1 (0.8660254 i)}{1}$

Step 2.8

Factor $1$ out of $1 (0.5) + 1 (0.8660254 i)$ .

$- 21 + 20 i + 3 + 4 i + \frac{1 (0.5 + 0.8660254 i)}{1}$

Step 2.9

Factor $1$ out of $1$ .

$- 21 + 20 i + 3 + 4 i + \frac{1 (0.5 + 0.8660254 i)}{1 (1)}$

Step 2.10

Separate fractions.

$- 21 + 20 i + 3 + 4 i + \frac{1}{1} \cdot \frac{0.5 + 0.8660254 i}{1}$

Step 2.11

Divide $1$ by $1$ .

$- 21 + 20 i + 3 + 4 i + 1 \frac{0.5 + 0.8660254 i}{1}$

Step 2.12

Divide $0.5 + 0.8660254 i$ by $1$ .

$- 21 + 20 i + 3 + 4 i + 1 (0.5 + 0.8660254 i)$

Step 2.13

Apply the distributive property.

$- 21 + 20 i + 3 + 4 i + 1 \cdot 0.5 + 1 (0.8660254 i)$

Step 2.14

Multiply $1$ by $0.5$ .

$- 21 + 20 i + 3 + 4 i + 0.5 + 1 (0.8660254 i)$

Step 2.15

Multiply $0.8660254$ by $1$ .

$- 21 + 20 i + 3 + 4 i + 0.5 + 0.8660254 i$

Step 3

Simplify by adding terms.

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Step 3.1

Add $- 21$ and $3$ .

$- 18 + 20 i + 4 i + 0.5 + 0.8660254 i$

Step 3.2

Add $- 18$ and $0.5$ .

$- 17.5 + 20 i + 4 i + 0.8660254 i$

Step 3.3

Add $20 i$ and $4 i$ .

$- 17.5 + 24 i + 0.8660254 i$

Step 3.4

Add $24 i$ and $0.8660254 i$ .

$- 17.5 + 24.8660254 i$