Algebra Examples Complex Numbers and Vector Analysis

Simplify

$\frac{8}{3 + 7 i}$

Step 1

Multiply the numerator and denominator of $\frac{8}{3 + 7 i}$ by the conjugate of $3 + 7 i$ to make the denominator real.

$\frac{8}{3 + 7 i} \cdot \frac{3 - 7 i}{3 - 7 i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $8$ by $3$ .

$\frac{24 + 8 (- 7 i)}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.3

Multiply $- 7$ by $8$ .

$\frac{24 - 56 i}{(3 + 7 i) (3 - 7 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{24 - 56 i}{3 (3 - 7 i) + 7 i (3 - 7 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{24 - 56 i}{3 \cdot 3 + 3 (- 7 i) + 7 i (3 - 7 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{24 - 56 i}{3 \cdot 3 + 3 (- 7 i) + 7 i \cdot 3 + 7 i (- 7 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $3$ by $3$ .

$\frac{24 - 56 i}{9 + 3 (- 7 i) + 7 i \cdot 3 + 7 i (- 7 i)}$

Step 2.3.2.2

Multiply $- 7$ by $3$ .

$\frac{24 - 56 i}{9 - 21 i + 7 i \cdot 3 + 7 i (- 7 i)}$

Step 2.3.2.3

Multiply $3$ by $7$ .

$\frac{24 - 56 i}{9 - 21 i + 21 i + 7 i (- 7 i)}$

Step 2.3.2.4

Multiply $- 7$ by $7$ .

$\frac{24 - 56 i}{9 - 21 i + 21 i - 49 i i}$

Step 2.3.2.5

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

$\frac{24 - 56 i}{9 - 21 i + 21 i - 49 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{24 - 56 i}{9 - 21 i + 21 i - 49 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{24 - 56 i}{9 - 21 i + 21 i - 49 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{24 - 56 i}{9 - 21 i + 21 i - 49 i^{2}}$

Step 2.3.2.9

Add $- 21 i$ and $21 i$ .

$\frac{24 - 56 i}{9 + 0 - 49 i^{2}}$

Step 2.3.2.10

Add $9$ and $0$ .

$\frac{24 - 56 i}{9 - 49 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{24 - 56 i}{9 - 49 \cdot - 1}$

Step 2.3.3.2

Multiply $- 49$ by $- 1$ .

$\frac{24 - 56 i}{9 + 49}$

Step 2.3.4

Add $9$ and $49$ .

$\frac{24 - 56 i}{58}$

Step 3

Cancel the common factor of $24 - 56 i$ and $58$ .

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Step 3.1

Factor $2$ out of $24$ .

$\frac{2 (12) - 56 i}{58}$

Step 3.2

Factor $2$ out of $- 56 i$ .

$\frac{2 (12) + 2 (- 28 i)}{58}$

Step 3.3

Factor $2$ out of $2 (12) + 2 (- 28 i)$ .

$\frac{2 (12 - 28 i)}{58}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $58$ .

$\frac{2 (12 - 28 i)}{2 \cdot 29}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (12 - 28 i)}{2 \cdot 29}$

Step 3.4.3

Rewrite the expression.

$\frac{12 - 28 i}{29}$

Step 4

Split the fraction $\frac{12 - 28 i}{29}$ into two fractions.

$\frac{12}{29} + \frac{- 28 i}{29}$

Step 5

Move the negative in front of the fraction.

$\frac{12}{29} - \frac{28 i}{29}$

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