Algebra Examples

$\frac{(2 + 3 i) (4 - 7 i)}{(1 - i)}$

Step 1

Multiply the numerator and denominator of $\frac{(2 + 3 i) (4 - 7 i)}{1 - i}$ by the conjugate of $1 - i$ to make the denominator real.

$\frac{(2 + 3 i) (4 - 7 i)}{1 - i} \cdot \frac{1 + i}{1 + i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(2 + 3 i) (4 - 7 i) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(2 + 3 i) (4 - 7 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{(2 (4 - 7 i) + 3 i (4 - 7 i)) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{(2 \cdot 4 + 2 (- 7 i) + 3 i (4 - 7 i)) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{(2 \cdot 4 + 2 (- 7 i) + 3 i \cdot 4 + 3 i (- 7 i)) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $2$ by $4$ .

$\frac{(8 + 2 (- 7 i) + 3 i \cdot 4 + 3 i (- 7 i)) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.2

Multiply $- 7$ by $2$ .

$\frac{(8 - 14 i + 3 i \cdot 4 + 3 i (- 7 i)) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.3

Multiply $4$ by $3$ .

$\frac{(8 - 14 i + 12 i + 3 i (- 7 i)) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.4

Multiply $3 i (- 7 i)$ .

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Step 2.2.2.1.4.1

Multiply $- 7$ by $3$ .

$\frac{(8 - 14 i + 12 i - 21 i i) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.4.2

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

$\frac{(8 - 14 i + 12 i - 21 (i^{1} i)) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.4.3

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

$\frac{(8 - 14 i + 12 i - 21 (i^{1} i^{1})) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.4.4

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

$\frac{(8 - 14 i + 12 i - 21 i^{1 + 1}) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{(8 - 14 i + 12 i - 21 i^{2}) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.5

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

$\frac{(8 - 14 i + 12 i - 21 \cdot - 1) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.1.6

Multiply $- 21$ by $- 1$ .

$\frac{(8 - 14 i + 12 i + 21) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.2

Add $8$ and $21$ .

$\frac{(29 - 14 i + 12 i) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.2.3

Add $- 14 i$ and $12 i$ .

$\frac{(29 - 2 i) (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.3

Expand $(29 - 2 i) (1 + i)$ using the FOIL Method.

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Step 2.2.3.1

Apply the distributive property.

$\frac{29 (1 + i) - 2 i (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.3.2

Apply the distributive property.

$\frac{29 \cdot 1 + 29 i - 2 i (1 + i)}{(1 - i) (1 + i)}$

Step 2.2.3.3

Apply the distributive property.

$\frac{29 \cdot 1 + 29 i - 2 i \cdot 1 - 2 i i}{(1 - i) (1 + i)}$

Step 2.2.4

Simplify and combine like terms.

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Multiply $29$ by $1$ .

$\frac{29 + 29 i - 2 i \cdot 1 - 2 i i}{(1 - i) (1 + i)}$

Step 2.2.4.1.2

Multiply $- 2$ by $1$ .

$\frac{29 + 29 i - 2 i - 2 i i}{(1 - i) (1 + i)}$

Step 2.2.4.1.3

Multiply $- 2 i i$ .

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Step 2.2.4.1.3.1

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

$\frac{29 + 29 i - 2 i - 2 (i^{1} i)}{(1 - i) (1 + i)}$

Step 2.2.4.1.3.2

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

$\frac{29 + 29 i - 2 i - 2 (i^{1} i^{1})}{(1 - i) (1 + i)}$

Step 2.2.4.1.3.3

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

$\frac{29 + 29 i - 2 i - 2 i^{1 + 1}}{(1 - i) (1 + i)}$

Step 2.2.4.1.3.4

Add $1$ and $1$ .

$\frac{29 + 29 i - 2 i - 2 i^{2}}{(1 - i) (1 + i)}$

Step 2.2.4.1.4

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

$\frac{29 + 29 i - 2 i - 2 \cdot - 1}{(1 - i) (1 + i)}$

Step 2.2.4.1.5

Multiply $- 2$ by $- 1$ .

$\frac{29 + 29 i - 2 i + 2}{(1 - i) (1 + i)}$

Step 2.2.4.2

Add $29$ and $2$ .

$\frac{31 + 29 i - 2 i}{(1 - i) (1 + i)}$

Step 2.2.4.3

Subtract $2 i$ from $29 i$ .

$\frac{31 + 27 i}{(1 - i) (1 + i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(1 - i) (1 + i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{31 + 27 i}{1 (1 + i) - i (1 + i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{31 + 27 i}{1 \cdot 1 + 1 i - i (1 + i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{31 + 27 i}{1 \cdot 1 + 1 i - i \cdot 1 - i i}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $1$ by $1$ .

$\frac{31 + 27 i}{1 + 1 i - i \cdot 1 - i i}$

Step 2.3.2.2

Multiply $- 1$ by $1$ .

$\frac{31 + 27 i}{1 + 1 i - i - i i}$

Step 2.3.2.3

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

$\frac{31 + 27 i}{1 + 1 i - i - (i^{1} i)}$

Step 2.3.2.4

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

$\frac{31 + 27 i}{1 + 1 i - i - (i^{1} i^{1})}$

Step 2.3.2.5

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

$\frac{31 + 27 i}{1 + 1 i - i - i^{1 + 1}}$

Step 2.3.2.6

Add $1$ and $1$ .

$\frac{31 + 27 i}{1 + 1 i - i - i^{2}}$

Step 2.3.2.7

Subtract $i$ from $1 i$ .

$\frac{31 + 27 i}{1 + 0 - i^{2}}$

Step 2.3.2.8

Add $1$ and $0$ .

$\frac{31 + 27 i}{1 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{31 + 27 i}{1 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{31 + 27 i}{1 + 1}$

Step 2.3.4

Add $1$ and $1$ .

$\frac{31 + 27 i}{2}$

Step 3

Split the fraction $\frac{31 + 27 i}{2}$ into two fractions.

$\frac{31}{2} + \frac{27 i}{2}$