Basic Math Examples

$\frac{- 26 + 36 i}{- 40 + 42 i}$

Step 1

Cancel the common factor of $- 26 + 36 i$ and $- 40 + 42 i$ .

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Step 1.1

Factor $2$ out of $- 26$ .

$\frac{2 (- 13) + 36 i}{- 40 + 42 i}$

Step 1.2

Factor $2$ out of $36 i$ .

$\frac{2 (- 13) + 2 (18 i)}{- 40 + 42 i}$

Step 1.3

Factor $2$ out of $2 (- 13) + 2 (18 i)$ .

$\frac{2 (- 13 + 18 i)}{- 40 + 42 i}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $- 40$ .

$\frac{2 (- 13 + 18 i)}{2 \cdot - 20 + 42 i}$

Step 1.4.2

Factor $2$ out of $42 i$ .

$\frac{2 (- 13 + 18 i)}{2 \cdot - 20 + 2 (21 i)}$

Step 1.4.3

Factor $2$ out of $2 (- 20) + 2 (21 i)$ .

$\frac{2 (- 13 + 18 i)}{2 (- 20 + 21 i)}$

Step 1.4.4

Cancel the common factor.

$\frac{2 (- 13 + 18 i)}{2 (- 20 + 21 i)}$

Step 1.4.5

Rewrite the expression.

$\frac{- 13 + 18 i}{- 20 + 21 i}$

Step 2

Multiply the numerator and denominator of $\frac{- 13 + 18 i}{- 20 + 21 i}$ by the conjugate of $- 20 + 21 i$ to make the denominator real.

$\frac{- 13 + 18 i}{- 20 + 21 i} \cdot \frac{- 20 - 21 i}{- 20 - 21 i}$

Step 3

Multiply.

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Step 3.1

Combine.

$\frac{(- 13 + 18 i) (- 20 - 21 i)}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Expand $(- 13 + 18 i) (- 20 - 21 i)$ using the FOIL Method.

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Step 3.2.1.1

Apply the distributive property.

$\frac{- 13 (- 20 - 21 i) + 18 i (- 20 - 21 i)}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.1.2

Apply the distributive property.

$\frac{- 13 \cdot - 20 - 13 (- 21 i) + 18 i (- 20 - 21 i)}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.1.3

Apply the distributive property.

$\frac{- 13 \cdot - 20 - 13 (- 21 i) + 18 i \cdot - 20 + 18 i (- 21 i)}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2

Simplify and combine like terms.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $- 13$ by $- 20$ .

$\frac{260 - 13 (- 21 i) + 18 i \cdot - 20 + 18 i (- 21 i)}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.2

Multiply $- 21$ by $- 13$ .

$\frac{260 + 273 i + 18 i \cdot - 20 + 18 i (- 21 i)}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.3

Multiply $- 20$ by $18$ .

$\frac{260 + 273 i - 360 i + 18 i (- 21 i)}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.4

Multiply $18 i (- 21 i)$ .

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Step 3.2.2.1.4.1

Multiply $- 21$ by $18$ .

$\frac{260 + 273 i - 360 i - 378 i i}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.4.2

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

$\frac{260 + 273 i - 360 i - 378 (i^{1} i)}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.4.3

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

$\frac{260 + 273 i - 360 i - 378 (i^{1} i^{1})}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.4.4

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

$\frac{260 + 273 i - 360 i - 378 i^{1 + 1}}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.4.5

Add $1$ and $1$ .

$\frac{260 + 273 i - 360 i - 378 i^{2}}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.5

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

$\frac{260 + 273 i - 360 i - 378 \cdot - 1}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.1.6

Multiply $- 378$ by $- 1$ .

$\frac{260 + 273 i - 360 i + 378}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.2

Add $260$ and $378$ .

$\frac{638 + 273 i - 360 i}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.2.2.3

Subtract $360 i$ from $273 i$ .

$\frac{638 - 87 i}{(- 20 + 21 i) (- 20 - 21 i)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand $(- 20 + 21 i) (- 20 - 21 i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$\frac{638 - 87 i}{- 20 (- 20 - 21 i) + 21 i (- 20 - 21 i)}$

Step 3.3.1.2

Apply the distributive property.

$\frac{638 - 87 i}{- 20 \cdot - 20 - 20 (- 21 i) + 21 i (- 20 - 21 i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{638 - 87 i}{- 20 \cdot - 20 - 20 (- 21 i) + 21 i \cdot - 20 + 21 i (- 21 i)}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $- 20$ by $- 20$ .

$\frac{638 - 87 i}{400 - 20 (- 21 i) + 21 i \cdot - 20 + 21 i (- 21 i)}$

Step 3.3.2.2

Multiply $- 21$ by $- 20$ .

$\frac{638 - 87 i}{400 + 420 i + 21 i \cdot - 20 + 21 i (- 21 i)}$

Step 3.3.2.3

Multiply $- 20$ by $21$ .

$\frac{638 - 87 i}{400 + 420 i - 420 i + 21 i (- 21 i)}$

Step 3.3.2.4

Multiply $- 21$ by $21$ .

$\frac{638 - 87 i}{400 + 420 i - 420 i - 441 i i}$

Step 3.3.2.5

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

$\frac{638 - 87 i}{400 + 420 i - 420 i - 441 (i^{1} i)}$

Step 3.3.2.6

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

$\frac{638 - 87 i}{400 + 420 i - 420 i - 441 (i^{1} i^{1})}$

Step 3.3.2.7

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

$\frac{638 - 87 i}{400 + 420 i - 420 i - 441 i^{1 + 1}}$

Step 3.3.2.8

Add $1$ and $1$ .

$\frac{638 - 87 i}{400 + 420 i - 420 i - 441 i^{2}}$

Step 3.3.2.9

Subtract $420 i$ from $420 i$ .

$\frac{638 - 87 i}{400 + 0 - 441 i^{2}}$

Step 3.3.2.10

Add $400$ and $0$ .

$\frac{638 - 87 i}{400 - 441 i^{2}}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

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

$\frac{638 - 87 i}{400 - 441 \cdot - 1}$

Step 3.3.3.2

Multiply $- 441$ by $- 1$ .

$\frac{638 - 87 i}{400 + 441}$

Step 3.3.4

Add $400$ and $441$ .

$\frac{638 - 87 i}{841}$

Step 4

Cancel the common factor of $638 - 87 i$ and $841$ .

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Step 4.1

Factor $29$ out of $638$ .

$\frac{29 (22) - 87 i}{841}$

Step 4.2

Factor $29$ out of $- 87 i$ .

$\frac{29 (22) + 29 (- 3 i)}{841}$

Step 4.3

Factor $29$ out of $29 (22) + 29 (- 3 i)$ .

$\frac{29 (22 - 3 i)}{841}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $29$ out of $841$ .

$\frac{29 (22 - 3 i)}{29 \cdot 29}$

Step 4.4.2

Cancel the common factor.

$\frac{29 (22 - 3 i)}{29 \cdot 29}$

Step 4.4.3

Rewrite the expression.

$\frac{22 - 3 i}{29}$

Step 5

Split the fraction $\frac{22 - 3 i}{29}$ into two fractions.

$\frac{22}{29} + \frac{- 3 i}{29}$

Step 6

Move the negative in front of the fraction.

$\frac{22}{29} - \frac{3 i}{29}$