Basic Math Examples

$6 - \frac{4 i}{2 - i}$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator and denominator of $\frac{4 i}{2 - i}$ by the conjugate of $2 - i$ to make the denominator real.

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

Step 1.2

Multiply.

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Step 1.2.1

Combine.

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

Step 1.2.2

Simplify the numerator.

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Multiply $2$ by $4$ .

$6 - \frac{8 i + 4 i i}{(2 - i) (2 + i)}$

Step 1.2.2.3

Multiply $4 i i$ .

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Step 1.2.2.3.1

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

$6 - \frac{8 i + 4 (i^{1} i)}{(2 - i) (2 + i)}$

Step 1.2.2.3.2

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

$6 - \frac{8 i + 4 (i^{1} i^{1})}{(2 - i) (2 + i)}$

Step 1.2.2.3.3

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

$6 - \frac{8 i + 4 i^{1 + 1}}{(2 - i) (2 + i)}$

Step 1.2.2.3.4

Add $1$ and $1$ .

$6 - \frac{8 i + 4 i^{2}}{(2 - i) (2 + i)}$

Step 1.2.2.4

Simplify each term.

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Step 1.2.2.4.1

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

$6 - \frac{8 i + 4 \cdot - 1}{(2 - i) (2 + i)}$

Step 1.2.2.4.2

Multiply $4$ by $- 1$ .

$6 - \frac{8 i - 4}{(2 - i) (2 + i)}$

Step 1.2.2.5

Reorder $8 i$ and $- 4$ .

$6 - \frac{- 4 + 8 i}{(2 - i) (2 + i)}$

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

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

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Step 1.2.3.1.1

Apply the distributive property.

$6 - \frac{- 4 + 8 i}{2 (2 + i) - i (2 + i)}$

Step 1.2.3.1.2

Apply the distributive property.

$6 - \frac{- 4 + 8 i}{2 \cdot 2 + 2 i - i (2 + i)}$

Step 1.2.3.1.3

Apply the distributive property.

$6 - \frac{- 4 + 8 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Multiply $2$ by $2$ .

$6 - \frac{- 4 + 8 i}{4 + 2 i - i \cdot 2 - i i}$

Step 1.2.3.2.2

Multiply $2$ by $- 1$ .

$6 - \frac{- 4 + 8 i}{4 + 2 i - 2 i - i i}$

Step 1.2.3.2.3

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

$6 - \frac{- 4 + 8 i}{4 + 2 i - 2 i - (i^{1} i)}$

Step 1.2.3.2.4

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

$6 - \frac{- 4 + 8 i}{4 + 2 i - 2 i - (i^{1} i^{1})}$

Step 1.2.3.2.5

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

$6 - \frac{- 4 + 8 i}{4 + 2 i - 2 i - i^{1 + 1}}$

Step 1.2.3.2.6

Add $1$ and $1$ .

$6 - \frac{- 4 + 8 i}{4 + 2 i - 2 i - i^{2}}$

Step 1.2.3.2.7

Subtract $2 i$ from $2 i$ .

$6 - \frac{- 4 + 8 i}{4 + 0 - i^{2}}$

Step 1.2.3.2.8

Add $4$ and $0$ .

$6 - \frac{- 4 + 8 i}{4 - i^{2}}$

Step 1.2.3.3

Simplify each term.

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Step 1.2.3.3.1

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

$6 - \frac{- 4 + 8 i}{4 - - 1}$

Step 1.2.3.3.2

Multiply $- 1$ by $- 1$ .

$6 - \frac{- 4 + 8 i}{4 + 1}$

Step 1.2.3.4

Add $4$ and $1$ .

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

Step 1.3

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

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

Step 1.4

Move the negative in front of the fraction.

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $- - \frac{4}{5}$ .

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Step 1.6.1

Multiply $- 1$ by $- 1$ .

$6 + 1 (\frac{4}{5}) - \frac{8 i}{5}$

Step 1.6.2

Multiply $\frac{4}{5}$ by $1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $6$ by $5$ .

$\frac{30 + 4}{5} - \frac{8 i}{5}$

Step 5.2

Add $30$ and $4$ .

$\frac{34}{5} - \frac{8 i}{5}$