Algebra Examples

$(\frac{3}{4} + \frac{7}{5} i) - (\frac{5}{6} - \frac{1}{6} i)$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{7}{5}$ and $i$ .

$\frac{3}{4} + \frac{7 i}{5} - (\frac{5}{6} - \frac{1}{6} i)$

Step 1.2

Combine $i$ and $\frac{1}{6}$ .

$\frac{3}{4} + \frac{7 i}{5} - (\frac{5}{6} - \frac{i}{6})$

Step 1.3

Apply the distributive property.

$\frac{3}{4} + \frac{7 i}{5} - \frac{5}{6} - - \frac{i}{6}$

Step 1.4

Multiply $- - \frac{i}{6}$ .

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Step 1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{4} + \frac{7 i}{5} - \frac{5}{6} + 1 \frac{i}{6}$

Step 1.4.2

Multiply $\frac{i}{6}$ by $1$ .

$\frac{3}{4} + \frac{7 i}{5} - \frac{5}{6} + \frac{i}{6}$

Step 2

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3}{4} \cdot \frac{3}{3} - \frac{5}{6} + \frac{7 i}{5} + \frac{i}{6}$

Step 3

To write $- \frac{5}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{4} \cdot \frac{3}{3} - \frac{5}{6} \cdot \frac{2}{2} + \frac{7 i}{5} + \frac{i}{6}$

Step 4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

$\frac{3 \cdot 3}{4 \cdot 3} - \frac{5}{6} \cdot \frac{2}{2} + \frac{7 i}{5} + \frac{i}{6}$

Step 4.2

Multiply $4$ by $3$ .

$\frac{3 \cdot 3}{12} - \frac{5}{6} \cdot \frac{2}{2} + \frac{7 i}{5} + \frac{i}{6}$

Step 4.3

Multiply $\frac{5}{6}$ by $\frac{2}{2}$ .

$\frac{3 \cdot 3}{12} - \frac{5 \cdot 2}{6 \cdot 2} + \frac{7 i}{5} + \frac{i}{6}$

Step 4.4

Multiply $6$ by $2$ .

$\frac{3 \cdot 3}{12} - \frac{5 \cdot 2}{12} + \frac{7 i}{5} + \frac{i}{6}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 - 5 \cdot 2}{12} + \frac{7 i}{5} + \frac{i}{6}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $3$ by $3$ .

$\frac{9 - 5 \cdot 2}{12} + \frac{7 i}{5} + \frac{i}{6}$

Step 6.2

Multiply $- 5$ by $2$ .

$\frac{9 - 10}{12} + \frac{7 i}{5} + \frac{i}{6}$

Step 6.3

Subtract $10$ from $9$ .

$\frac{- 1}{12} + \frac{7 i}{5} + \frac{i}{6}$

Step 7

Move the negative in front of the fraction.

$- \frac{1}{12} + \frac{7 i}{5} + \frac{i}{6}$

Step 8

To write $\frac{7 i}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{1}{12} + \frac{7 i}{5} \cdot \frac{6}{6} + \frac{i}{6}$

Step 9

To write $\frac{i}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{1}{12} + \frac{7 i}{5} \cdot \frac{6}{6} + \frac{i}{6} \cdot \frac{5}{5}$

Step 10

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{7 i}{5}$ by $\frac{6}{6}$ .

$- \frac{1}{12} + \frac{7 i \cdot 6}{5 \cdot 6} + \frac{i}{6} \cdot \frac{5}{5}$

Step 10.2

Multiply $5$ by $6$ .

$- \frac{1}{12} + \frac{7 i \cdot 6}{30} + \frac{i}{6} \cdot \frac{5}{5}$

Step 10.3

Multiply $\frac{i}{6}$ by $\frac{5}{5}$ .

$- \frac{1}{12} + \frac{7 i \cdot 6}{30} + \frac{i \cdot 5}{6 \cdot 5}$

Step 10.4

Multiply $6$ by $5$ .

$- \frac{1}{12} + \frac{7 i \cdot 6}{30} + \frac{i \cdot 5}{30}$

Step 11

Combine the numerators over the common denominator.

$- \frac{1}{12} + \frac{47 i}{30}$