Algebra Examples

$\frac{3 - 4 i}{5} (\frac{4}{5} + \frac{3 i}{5})$

Step 1

Split the fraction $\frac{3 - 4 i}{5}$ into two fractions.

$(\frac{3}{5} + \frac{- 4 i}{5}) \cdot (\frac{4}{5} + \frac{3 i}{5})$

Step 2

Move the negative in front of the fraction.

$(\frac{3}{5} - \frac{4 i}{5}) \cdot (\frac{4}{5} + \frac{3 i}{5})$

Step 3

Expand $(\frac{3}{5} - \frac{4 i}{5}) (\frac{4}{5} + \frac{3 i}{5})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{3}{5} (\frac{4}{5} + \frac{3 i}{5}) - \frac{4 i}{5} (\frac{4}{5} + \frac{3 i}{5})$

Step 3.2

Apply the distributive property.

$\frac{3}{5} \cdot \frac{4}{5} + \frac{3}{5} \cdot \frac{3 i}{5} - \frac{4 i}{5} (\frac{4}{5} + \frac{3 i}{5})$

Step 3.3

Apply the distributive property.

$\frac{3}{5} \cdot \frac{4}{5} + \frac{3}{5} \cdot \frac{3 i}{5} - \frac{4 i}{5} \cdot \frac{4}{5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $\frac{3}{5} \cdot \frac{4}{5}$ .

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Step 4.1.1.1

Multiply $\frac{3}{5}$ by $\frac{4}{5}$ .

$\frac{3 \cdot 4}{5 \cdot 5} + \frac{3}{5} \cdot \frac{3 i}{5} - \frac{4 i}{5} \cdot \frac{4}{5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.1.2

Multiply $3$ by $4$ .

$\frac{12}{5 \cdot 5} + \frac{3}{5} \cdot \frac{3 i}{5} - \frac{4 i}{5} \cdot \frac{4}{5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.1.3

Multiply $5$ by $5$ .

$\frac{12}{25} + \frac{3}{5} \cdot \frac{3 i}{5} - \frac{4 i}{5} \cdot \frac{4}{5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.2

Multiply $\frac{3}{5} \cdot \frac{3 i}{5}$ .

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Step 4.1.2.1

Multiply $\frac{3}{5}$ by $\frac{3 i}{5}$ .

$\frac{12}{25} + \frac{3 (3 i)}{5 \cdot 5} - \frac{4 i}{5} \cdot \frac{4}{5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.2.2

Multiply $3$ by $3$ .

$\frac{12}{25} + \frac{9 i}{5 \cdot 5} - \frac{4 i}{5} \cdot \frac{4}{5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.2.3

Multiply $5$ by $5$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{4 i}{5} \cdot \frac{4}{5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.3

Multiply $- \frac{4 i}{5} \cdot \frac{4}{5}$ .

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Step 4.1.3.1

Multiply $\frac{4}{5}$ by $\frac{4 i}{5}$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{4 (4 i)}{5 \cdot 5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.3.2

Multiply $4$ by $4$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{5 \cdot 5} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.3.3

Multiply $5$ by $5$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{4 i}{5} \cdot \frac{3 i}{5}$

Step 4.1.4

Multiply $- \frac{4 i}{5} \cdot \frac{3 i}{5}$ .

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Step 4.1.4.1

Multiply $\frac{3 i}{5}$ by $\frac{4 i}{5}$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{3 i (4 i)}{5 \cdot 5}$

Step 4.1.4.2

Multiply $4$ by $3$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{12 i i}{5 \cdot 5}$

Step 4.1.4.3

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

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{12 (i^{1} i)}{5 \cdot 5}$

Step 4.1.4.4

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

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{12 (i^{1} i^{1})}{5 \cdot 5}$

Step 4.1.4.5

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

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{12 i^{1 + 1}}{5 \cdot 5}$

Step 4.1.4.6

Add $1$ and $1$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{12 i^{2}}{5 \cdot 5}$

Step 4.1.4.7

Multiply $5$ by $5$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{12 i^{2}}{25}$

Step 4.1.5

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

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{12 \cdot - 1}{25}$

Step 4.1.6

Multiply $12$ by $- 1$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - \frac{- 12}{25}$

Step 4.1.7

Move the negative in front of the fraction.

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} - - \frac{12}{25}$

Step 4.1.8

Multiply $- - \frac{12}{25}$ .

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Step 4.1.8.1

Multiply $- 1$ by $- 1$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} + 1 (\frac{12}{25})$

Step 4.1.8.2

Multiply $\frac{12}{25}$ by $1$ .

$\frac{12}{25} + \frac{9 i}{25} - \frac{16 i}{25} + \frac{12}{25}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{12 + 12}{25} + \frac{9 i}{25} - \frac{16 i}{25}$

Step 4.3

Add $12$ and $12$ .

$\frac{24}{25} + \frac{9 i}{25} - \frac{16 i}{25}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{24}{25} + \frac{- 7 i}{25}$

Step 5

Move the negative in front of the fraction.

$\frac{24}{25} - \frac{7 i}{25}$