Algebra Examples

$\frac{8 - 7 i}{1 - 2 i}$

Step 1

Multiply the numerator and denominator of $\frac{8 - 7 i}{1 - 2 i}$ by the conjugate of $1 - 2 i$ to make the denominator real.

$\frac{8 - 7 i}{1 - 2 i} \cdot \frac{1 + 2 i}{1 + 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(8 - 7 i) (1 + 2 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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 $8$ by $1$ .

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

Step 2.2.2.1.2

Multiply $2$ by $8$ .

$\frac{8 + 16 i - 7 i \cdot 1 - 7 i (2 i)}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.1.3

Multiply $- 7$ by $1$ .

$\frac{8 + 16 i - 7 i - 7 i (2 i)}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.1.4

Multiply $- 7 i (2 i)$ .

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Step 2.2.2.1.4.1

Multiply $2$ by $- 7$ .

$\frac{8 + 16 i - 7 i - 14 i i}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.1.4.2

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

$\frac{8 + 16 i - 7 i - 14 (i^{1} i)}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.1.4.3

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

$\frac{8 + 16 i - 7 i - 14 (i^{1} i^{1})}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.1.4.4

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

$\frac{8 + 16 i - 7 i - 14 i^{1 + 1}}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{8 + 16 i - 7 i - 14 i^{2}}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.1.5

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

$\frac{8 + 16 i - 7 i - 14 \cdot - 1}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.1.6

Multiply $- 14$ by $- 1$ .

$\frac{8 + 16 i - 7 i + 14}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.2

Add $8$ and $14$ .

$\frac{22 + 16 i - 7 i}{(1 - 2 i) (1 + 2 i)}$

Step 2.2.2.3

Subtract $7 i$ from $16 i$ .

$\frac{22 + 9 i}{(1 - 2 i) (1 + 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{22 + 9 i}{1 (1 + 2 i) - 2 i (1 + 2 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{22 + 9 i}{1 \cdot 1 + 1 (2 i) - 2 i (1 + 2 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{22 + 9 i}{1 \cdot 1 + 1 (2 i) - 2 i \cdot 1 - 2 i (2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $1$ by $1$ .

$\frac{22 + 9 i}{1 + 1 (2 i) - 2 i \cdot 1 - 2 i (2 i)}$

Step 2.3.2.2

Multiply $2$ by $1$ .

$\frac{22 + 9 i}{1 + 2 i - 2 i \cdot 1 - 2 i (2 i)}$

Step 2.3.2.3

Multiply $- 2$ by $1$ .

$\frac{22 + 9 i}{1 + 2 i - 2 i - 2 i (2 i)}$

Step 2.3.2.4

Multiply $2$ by $- 2$ .

$\frac{22 + 9 i}{1 + 2 i - 2 i - 4 i i}$

Step 2.3.2.5

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

$\frac{22 + 9 i}{1 + 2 i - 2 i - 4 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{22 + 9 i}{1 + 2 i - 2 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{22 + 9 i}{1 + 2 i - 2 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{22 + 9 i}{1 + 2 i - 2 i - 4 i^{2}}$

Step 2.3.2.9

Subtract $2 i$ from $2 i$ .

$\frac{22 + 9 i}{1 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add $1$ and $0$ .

$\frac{22 + 9 i}{1 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{22 + 9 i}{1 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{22 + 9 i}{1 + 4}$

Step 2.3.4

Add $1$ and $4$ .

$\frac{22 + 9 i}{5}$

Step 3

Split the fraction $\frac{22 + 9 i}{5}$ into two fractions.

$\frac{22}{5} + \frac{9 i}{5}$