Algebra Examples

$\frac{(5 + i) (6 - 5 i)}{2 i (- 3 + 7 i)}$

Step 1

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

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Step 1.1

Apply the distributive property.

$\frac{5 (6 - 5 i) + i (6 - 5 i)}{2 i (- 3 + 7 i)}$

Step 1.2

Apply the distributive property.

$\frac{5 \cdot 6 + 5 (- 5 i) + i (6 - 5 i)}{2 i (- 3 + 7 i)}$

Step 1.3

Apply the distributive property.

$\frac{5 \cdot 6 + 5 (- 5 i) + i \cdot 6 + i (- 5 i)}{2 i (- 3 + 7 i)}$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $5$ by $6$ .

$\frac{30 + 5 (- 5 i) + i \cdot 6 + i (- 5 i)}{2 i (- 3 + 7 i)}$

Step 2.1.2

Multiply $- 5$ by $5$ .

$\frac{30 - 25 i + i \cdot 6 + i (- 5 i)}{2 i (- 3 + 7 i)}$

Step 2.1.3

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

$\frac{30 - 25 i + 6 \cdot i + i (- 5 i)}{2 i (- 3 + 7 i)}$

Step 2.1.4

Multiply $i (- 5 i)$ .

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Step 2.1.4.1

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

$\frac{30 - 25 i + 6 i - 5 (i^{1} i)}{2 i (- 3 + 7 i)}$

Step 2.1.4.2

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

$\frac{30 - 25 i + 6 i - 5 (i^{1} i^{1})}{2 i (- 3 + 7 i)}$

Step 2.1.4.3

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

$\frac{30 - 25 i + 6 i - 5 i^{1 + 1}}{2 i (- 3 + 7 i)}$

Step 2.1.4.4

Add $1$ and $1$ .

$\frac{30 - 25 i + 6 i - 5 i^{2}}{2 i (- 3 + 7 i)}$

Step 2.1.5

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

$\frac{30 - 25 i + 6 i - 5 \cdot - 1}{2 i (- 3 + 7 i)}$

Step 2.1.6

Multiply $- 5$ by $- 1$ .

$\frac{30 - 25 i + 6 i + 5}{2 i (- 3 + 7 i)}$

Step 2.2

Add $30$ and $5$ .

$\frac{35 - 25 i + 6 i}{2 i (- 3 + 7 i)}$

Step 2.3

Add $- 25 i$ and $6 i$ .

$\frac{35 - 19 i}{2 i (- 3 + 7 i)}$

Step 3

Group $- 3 + 7 i$ and $i$ together.

$\frac{35 - 19 i}{2 ((- 3 + 7 i) i)}$

Step 4

Apply the distributive property.

$\frac{35 - 19 i}{2 (- 3 i + 7 i i)}$

Step 5

Multiply $7 i i$ .

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Step 5.1

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

$\frac{35 - 19 i}{2 (- 3 i + 7 (i^{1} i))}$

Step 5.2

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

$\frac{35 - 19 i}{2 (- 3 i + 7 (i^{1} i^{1}))}$

Step 5.3

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

$\frac{35 - 19 i}{2 (- 3 i + 7 i^{1 + 1})}$

Step 5.4

Add $1$ and $1$ .

$\frac{35 - 19 i}{2 (- 3 i + 7 i^{2})}$

Step 6

Simplify each term.

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Step 6.1

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

$\frac{35 - 19 i}{2 (- 3 i + 7 \cdot - 1)}$

Step 6.2

Multiply $7$ by $- 1$ .

$\frac{35 - 19 i}{2 (- 3 i - 7)}$

Step 7

Reorder $- 3 i$ and $- 7$ .

$\frac{35 - 19 i}{2 (- 7 - 3 i)}$

Step 8

Multiply the numerator and denominator of $\frac{35 - 19 i}{- 14 - 6 i}$ by the conjugate of $- 14 - 6 i$ to make the denominator real.

$\frac{35 - 19 i}{- 14 - 6 i} \cdot \frac{- 14 + 6 i}{- 14 + 6 i}$

Step 9

Multiply.

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Step 9.1

Combine.

$\frac{(35 - 19 i) (- 14 + 6 i)}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2

Simplify the numerator.

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Step 9.2.1

Expand $(35 - 19 i) (- 14 + 6 i)$ using the FOIL Method.

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Step 9.2.1.1

Apply the distributive property.

$\frac{35 (- 14 + 6 i) - 19 i (- 14 + 6 i)}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.1.2

Apply the distributive property.

$\frac{35 \cdot - 14 + 35 (6 i) - 19 i (- 14 + 6 i)}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.1.3

Apply the distributive property.

$\frac{35 \cdot - 14 + 35 (6 i) - 19 i \cdot - 14 - 19 i (6 i)}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2

Simplify and combine like terms.

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Step 9.2.2.1

Simplify each term.

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Step 9.2.2.1.1

Multiply $35$ by $- 14$ .

$\frac{- 490 + 35 (6 i) - 19 i \cdot - 14 - 19 i (6 i)}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.2

Multiply $6$ by $35$ .

$\frac{- 490 + 210 i - 19 i \cdot - 14 - 19 i (6 i)}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.3

Multiply $- 14$ by $- 19$ .

$\frac{- 490 + 210 i + 266 i - 19 i (6 i)}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.4

Multiply $- 19 i (6 i)$ .

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Step 9.2.2.1.4.1

Multiply $6$ by $- 19$ .

$\frac{- 490 + 210 i + 266 i - 114 i i}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.4.2

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

$\frac{- 490 + 210 i + 266 i - 114 (i^{1} i)}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.4.3

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

$\frac{- 490 + 210 i + 266 i - 114 (i^{1} i^{1})}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.4.4

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

$\frac{- 490 + 210 i + 266 i - 114 i^{1 + 1}}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.4.5

Add $1$ and $1$ .

$\frac{- 490 + 210 i + 266 i - 114 i^{2}}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.5

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

$\frac{- 490 + 210 i + 266 i - 114 \cdot - 1}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.1.6

Multiply $- 114$ by $- 1$ .

$\frac{- 490 + 210 i + 266 i + 114}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.2

Add $- 490$ and $114$ .

$\frac{- 376 + 210 i + 266 i}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.2.2.3

Add $210 i$ and $266 i$ .

$\frac{- 376 + 476 i}{(- 14 - 6 i) (- 14 + 6 i)}$

Step 9.3

Simplify the denominator.

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Step 9.3.1

Expand $(- 14 - 6 i) (- 14 + 6 i)$ using the FOIL Method.

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Step 9.3.1.1

Apply the distributive property.

$\frac{- 376 + 476 i}{- 14 (- 14 + 6 i) - 6 i (- 14 + 6 i)}$

Step 9.3.1.2

Apply the distributive property.

$\frac{- 376 + 476 i}{- 14 \cdot - 14 - 14 (6 i) - 6 i (- 14 + 6 i)}$

Step 9.3.1.3

Apply the distributive property.

$\frac{- 376 + 476 i}{- 14 \cdot - 14 - 14 (6 i) - 6 i \cdot - 14 - 6 i (6 i)}$

Step 9.3.2

Simplify.

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Step 9.3.2.1

Multiply $- 14$ by $- 14$ .

$\frac{- 376 + 476 i}{196 - 14 (6 i) - 6 i \cdot - 14 - 6 i (6 i)}$

Step 9.3.2.2

Multiply $6$ by $- 14$ .

$\frac{- 376 + 476 i}{196 - 84 i - 6 i \cdot - 14 - 6 i (6 i)}$

Step 9.3.2.3

Multiply $- 14$ by $- 6$ .

$\frac{- 376 + 476 i}{196 - 84 i + 84 i - 6 i (6 i)}$

Step 9.3.2.4

Multiply $6$ by $- 6$ .

$\frac{- 376 + 476 i}{196 - 84 i + 84 i - 36 i i}$

Step 9.3.2.5

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

$\frac{- 376 + 476 i}{196 - 84 i + 84 i - 36 (i^{1} i)}$

Step 9.3.2.6

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

$\frac{- 376 + 476 i}{196 - 84 i + 84 i - 36 (i^{1} i^{1})}$

Step 9.3.2.7

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

$\frac{- 376 + 476 i}{196 - 84 i + 84 i - 36 i^{1 + 1}}$

Step 9.3.2.8

Add $1$ and $1$ .

$\frac{- 376 + 476 i}{196 - 84 i + 84 i - 36 i^{2}}$

Step 9.3.2.9

Add $- 84 i$ and $84 i$ .

$\frac{- 376 + 476 i}{196 + 0 - 36 i^{2}}$

Step 9.3.2.10

Add $196$ and $0$ .

$\frac{- 376 + 476 i}{196 - 36 i^{2}}$

Step 9.3.3

Simplify each term.

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Step 9.3.3.1

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

$\frac{- 376 + 476 i}{196 - 36 \cdot - 1}$

Step 9.3.3.2

Multiply $- 36$ by $- 1$ .

$\frac{- 376 + 476 i}{196 + 36}$

Step 9.3.4

Add $196$ and $36$ .

$\frac{- 376 + 476 i}{232}$

Step 10

Cancel the common factor of $- 376 + 476 i$ and $232$ .

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Step 10.1

Factor $4$ out of $- 376$ .

$\frac{4 (- 94) + 476 i}{232}$

Step 10.2

Factor $4$ out of $476 i$ .

$\frac{4 (- 94) + 4 (119 i)}{232}$

Step 10.3

Factor $4$ out of $4 (- 94) + 4 (119 i)$ .

$\frac{4 (- 94 + 119 i)}{232}$

Step 10.4

Cancel the common factors.

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Step 10.4.1

Factor $4$ out of $232$ .

$\frac{4 (- 94 + 119 i)}{4 \cdot 58}$

Step 10.4.2

Cancel the common factor.

$\frac{4 (- 94 + 119 i)}{4 \cdot 58}$

Step 10.4.3

Rewrite the expression.

$\frac{- 94 + 119 i}{58}$

Step 11

Split the fraction $\frac{- 94 + 119 i}{58}$ into two fractions.

$\frac{- 94}{58} + \frac{119 i}{58}$

Step 12

Simplify each term.

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Step 12.1

Cancel the common factor of $- 94$ and $58$ .

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Step 12.1.1

Factor $2$ out of $- 94$ .

$\frac{2 (- 47)}{58} + \frac{119 i}{58}$

Step 12.1.2

Cancel the common factors.

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Step 12.1.2.1

Factor $2$ out of $58$ .

$\frac{2 \cdot - 47}{2 \cdot 29} + \frac{119 i}{58}$

Step 12.1.2.2

Cancel the common factor.

$\frac{2 \cdot - 47}{2 \cdot 29} + \frac{119 i}{58}$

Step 12.1.2.3

Rewrite the expression.

$\frac{- 47}{29} + \frac{119 i}{58}$

Step 12.2

Move the negative in front of the fraction.

$- \frac{47}{29} + \frac{119 i}{58}$