Finite Math Examples

$i (- 6 + \frac{11}{5} \cdot i) + \frac{2 + i}{2 - i}$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{11}{5}$ and $i$ .

$i (- 6 + \frac{11 i}{5}) + \frac{2 + i}{2 - i}$

Step 1.2

Apply the distributive property.

$i \cdot - 6 + i \frac{11 i}{5} + \frac{2 + i}{2 - i}$

Step 1.3

Move $- 6$ to the left of $i$ .

$- 6 \cdot i + i \frac{11 i}{5} + \frac{2 + i}{2 - i}$

Step 1.4

Multiply $i \frac{11 i}{5}$ .

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Step 1.4.1

Combine $i$ and $\frac{11 i}{5}$ .

$- 6 \cdot i + \frac{i (11 i)}{5} + \frac{2 + i}{2 - i}$

Step 1.4.2

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

$- 6 \cdot i + \frac{11 (i^{1} i)}{5} + \frac{2 + i}{2 - i}$

Step 1.4.3

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

$- 6 \cdot i + \frac{11 (i^{1} i^{1})}{5} + \frac{2 + i}{2 - i}$

Step 1.4.4

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

$- 6 \cdot i + \frac{11 i^{1 + 1}}{5} + \frac{2 + i}{2 - i}$

Step 1.4.5

Add $1$ and $1$ .

$- 6 \cdot i + \frac{11 i^{2}}{5} + \frac{2 + i}{2 - i}$

Step 1.5

Simplify each term.

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Step 1.5.1

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

$- 6 i + \frac{11 \cdot - 1}{5} + \frac{2 + i}{2 - i}$

Step 1.5.2

Multiply $11$ by $- 1$ .

$- 6 i + \frac{- 11}{5} + \frac{2 + i}{2 - i}$

Step 1.5.3

Move the negative in front of the fraction.

$- 6 i - \frac{11}{5} + \frac{2 + i}{2 - i}$

Step 1.6

Multiply the numerator and denominator of $\frac{2 + i}{2 - i}$ by the conjugate of $2 - i$ to make the denominator real.

$- 6 i - \frac{11}{5} + \frac{2 + i}{2 - i} \cdot \frac{2 + i}{2 + i}$

Step 1.7

Multiply.

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Step 1.7.1

Combine.

$- 6 i - \frac{11}{5} + \frac{(2 + i) (2 + i)}{(2 - i) (2 + i)}$

Step 1.7.2

Simplify the numerator.

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Step 1.7.2.1

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

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Step 1.7.2.1.1

Apply the distributive property.

$- 6 i - \frac{11}{5} + \frac{2 (2 + i) + i (2 + i)}{(2 - i) (2 + i)}$

Step 1.7.2.1.2

Apply the distributive property.

$- 6 i - \frac{11}{5} + \frac{2 \cdot 2 + 2 i + i (2 + i)}{(2 - i) (2 + i)}$

Step 1.7.2.1.3

Apply the distributive property.

$- 6 i - \frac{11}{5} + \frac{2 \cdot 2 + 2 i + i \cdot 2 + i i}{(2 - i) (2 + i)}$

Step 1.7.2.2

Simplify and combine like terms.

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Step 1.7.2.2.1

Simplify each term.

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Step 1.7.2.2.1.1

Multiply $2$ by $2$ .

$- 6 i - \frac{11}{5} + \frac{4 + 2 i + i \cdot 2 + i i}{(2 - i) (2 + i)}$

Step 1.7.2.2.1.2

Move $2$ to the left of $i$ .

$- 6 i - \frac{11}{5} + \frac{4 + 2 i + 2 \cdot i + i i}{(2 - i) (2 + i)}$

Step 1.7.2.2.1.3

Multiply $i i$ .

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Step 1.7.2.2.1.3.1

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

$- 6 i - \frac{11}{5} + \frac{4 + 2 i + 2 i + i^{1} i}{(2 - i) (2 + i)}$

Step 1.7.2.2.1.3.2

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

$- 6 i - \frac{11}{5} + \frac{4 + 2 i + 2 i + i^{1} i^{1}}{(2 - i) (2 + i)}$

Step 1.7.2.2.1.3.3

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

$- 6 i - \frac{11}{5} + \frac{4 + 2 i + 2 i + i^{1 + 1}}{(2 - i) (2 + i)}$

Step 1.7.2.2.1.3.4

Add $1$ and $1$ .

$- 6 i - \frac{11}{5} + \frac{4 + 2 i + 2 i + i^{2}}{(2 - i) (2 + i)}$

Step 1.7.2.2.1.4

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

$- 6 i - \frac{11}{5} + \frac{4 + 2 i + 2 i - 1}{(2 - i) (2 + i)}$

Step 1.7.2.2.2

Subtract $1$ from $4$ .

$- 6 i - \frac{11}{5} + \frac{3 + 2 i + 2 i}{(2 - i) (2 + i)}$

Step 1.7.2.2.3

Add $2 i$ and $2 i$ .

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{(2 - i) (2 + i)}$

Step 1.7.3

Simplify the denominator.

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Step 1.7.3.1

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

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Step 1.7.3.1.1

Apply the distributive property.

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{2 (2 + i) - i (2 + i)}$

Step 1.7.3.1.2

Apply the distributive property.

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{2 \cdot 2 + 2 i - i (2 + i)}$

Step 1.7.3.1.3

Apply the distributive property.

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}$

Step 1.7.3.2

Simplify.

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Step 1.7.3.2.1

Multiply $2$ by $2$ .

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 + 2 i - i \cdot 2 - i i}$

Step 1.7.3.2.2

Multiply $2$ by $- 1$ .

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 + 2 i - 2 i - i i}$

Step 1.7.3.2.3

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

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 + 2 i - 2 i - (i^{1} i)}$

Step 1.7.3.2.4

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

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 + 2 i - 2 i - (i^{1} i^{1})}$

Step 1.7.3.2.5

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

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 + 2 i - 2 i - i^{1 + 1}}$

Step 1.7.3.2.6

Add $1$ and $1$ .

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 + 2 i - 2 i - i^{2}}$

Step 1.7.3.2.7

Subtract $2 i$ from $2 i$ .

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 + 0 - i^{2}}$

Step 1.7.3.2.8

Add $4$ and $0$ .

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 - i^{2}}$

Step 1.7.3.3

Simplify each term.

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Step 1.7.3.3.1

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

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 - - 1}$

Step 1.7.3.3.2

Multiply $- 1$ by $- 1$ .

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{4 + 1}$

Step 1.7.3.4

Add $4$ and $1$ .

$- 6 i - \frac{11}{5} + \frac{3 + 4 i}{5}$

Step 1.8

Split the fraction $\frac{3 + 4 i}{5}$ into two fractions.

$- 6 i - \frac{11}{5} + \frac{3}{5} + \frac{4 i}{5}$

Step 2

Combine the numerators over the common denominator.

$- 6 i + \frac{- 11 + 3 + 4 i}{5}$

Step 3

Add $- 11$ and $3$ .

$- 6 i + \frac{- 8 + 4 i}{5}$

Step 4

Simplify each term.

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Step 4.1

Split the fraction $\frac{- 8 + 4 i}{5}$ into two fractions.

$- 6 i + \frac{- 8}{5} + \frac{4 i}{5}$

Step 4.2

Move the negative in front of the fraction.

$- 6 i - \frac{8}{5} + \frac{4 i}{5}$

Step 5

To write $- 6 i$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 6 i \cdot \frac{5}{5} + \frac{4 i}{5} - \frac{8}{5}$

Step 6

Combine $- 6 i$ and $\frac{5}{5}$ .

$\frac{- 6 i \cdot 5}{5} + \frac{4 i}{5} - \frac{8}{5}$

Step 7

Combine the numerators over the common denominator.

$\frac{- 26 i}{5} - \frac{8}{5}$

Step 8

Move the negative in front of the fraction.

$- \frac{26 i}{5} - \frac{8}{5}$

Step 9

Reorder $- \frac{26 i}{5}$ and $- \frac{8}{5}$ .

$- \frac{8}{5} - \frac{26 i}{5}$