Basic Math Examples

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

Step 1

Multiply the numerator and denominator of $\frac{2 i}{- 5 - 6 i}$ by the conjugate of $- 5 - 6 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $- 5$ by $2$ .

$\frac{- 10 i + 2 i (6 i)}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.2.3

Multiply $2 i (6 i)$ .

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Step 2.2.3.1

Multiply $6$ by $2$ .

$\frac{- 10 i + 12 i i}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.2.3.2

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

$\frac{- 10 i + 12 (i^{1} i)}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.2.3.3

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

$\frac{- 10 i + 12 (i^{1} i^{1})}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.2.3.4

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

$\frac{- 10 i + 12 i^{1 + 1}}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.2.3.5

Add $1$ and $1$ .

$\frac{- 10 i + 12 i^{2}}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

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

$\frac{- 10 i + 12 \cdot - 1}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.2.4.2

Multiply $12$ by $- 1$ .

$\frac{- 10 i - 12}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.2.5

Reorder $- 10 i$ and $- 12$ .

$\frac{- 12 - 10 i}{(- 5 - 6 i) (- 5 + 6 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(- 5 - 6 i) (- 5 + 6 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 12 - 10 i}{- 5 (- 5 + 6 i) - 6 i (- 5 + 6 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 12 - 10 i}{- 5 \cdot - 5 - 5 (6 i) - 6 i (- 5 + 6 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 12 - 10 i}{- 5 \cdot - 5 - 5 (6 i) - 6 i \cdot - 5 - 6 i (6 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $- 5$ by $- 5$ .

$\frac{- 12 - 10 i}{25 - 5 (6 i) - 6 i \cdot - 5 - 6 i (6 i)}$

Step 2.3.2.2

Multiply $6$ by $- 5$ .

$\frac{- 12 - 10 i}{25 - 30 i - 6 i \cdot - 5 - 6 i (6 i)}$

Step 2.3.2.3

Multiply $- 5$ by $- 6$ .

$\frac{- 12 - 10 i}{25 - 30 i + 30 i - 6 i (6 i)}$

Step 2.3.2.4

Multiply $6$ by $- 6$ .

$\frac{- 12 - 10 i}{25 - 30 i + 30 i - 36 i i}$

Step 2.3.2.5

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

$\frac{- 12 - 10 i}{25 - 30 i + 30 i - 36 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 12 - 10 i}{25 - 30 i + 30 i - 36 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 12 - 10 i}{25 - 30 i + 30 i - 36 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 12 - 10 i}{25 - 30 i + 30 i - 36 i^{2}}$

Step 2.3.2.9

Add $- 30 i$ and $30 i$ .

$\frac{- 12 - 10 i}{25 + 0 - 36 i^{2}}$

Step 2.3.2.10

Add $25$ and $0$ .

$\frac{- 12 - 10 i}{25 - 36 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 12 - 10 i}{25 - 36 \cdot - 1}$

Step 2.3.3.2

Multiply $- 36$ by $- 1$ .

$\frac{- 12 - 10 i}{25 + 36}$

Step 2.3.4

Add $25$ and $36$ .

$\frac{- 12 - 10 i}{61}$

Step 3

Split the fraction $\frac{- 12 - 10 i}{61}$ into two fractions.

$\frac{- 12}{61} + \frac{- 10 i}{61}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{12}{61} + \frac{- 10 i}{61}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{12}{61} - \frac{10 i}{61}$