Simplify (6/5+3/5i)(2/3-1/3i)

$(\frac{6}{5} + \frac{3}{5} \cdot i) (\frac{2}{3} - \frac{1}{3} \cdot i)$

Step 1

Simplify terms.

Combine $\frac{3}{5}$ and $i$ .

$(\frac{6}{5} + \frac{3 i}{5}) (\frac{2}{3} - \frac{1}{3} \cdot i)$

Combine $i$ and $\frac{1}{3}$ .

$(\frac{6}{5} + \frac{3 i}{5}) (\frac{2}{3} - \frac{i}{3})$

Step 2

Expand $(\frac{6}{5} + \frac{3 i}{5}) (\frac{2}{3} - \frac{i}{3})$ using the FOIL Method.

Apply the distributive property.

$\frac{6}{5} (\frac{2}{3} - \frac{i}{3}) + \frac{3 i}{5} (\frac{2}{3} - \frac{i}{3})$

Apply the distributive property.

$\frac{6}{5} \cdot \frac{2}{3} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} (\frac{2}{3} - \frac{i}{3})$

Apply the distributive property.

$\frac{6}{5} \cdot \frac{2}{3} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3

Simplify and combine like terms.

Simplify each term.

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$\frac{3 (2)}{5} \cdot \frac{2}{3} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Cancel the common factor.

$\frac{3 \cdot 2}{5} \cdot \frac{2}{3} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Rewrite the expression.

$\frac{2}{5} \cdot 2 + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Combine $\frac{2}{5}$ and $2$ .

$\frac{2 \cdot 2}{5} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Multiply $2$ by $2$ .

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{i}{3}$ into the numerator.

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

Factor $3$ out of $6$ .

$\frac{4}{5} + \frac{3 (2)}{5} \cdot \frac{- i}{3} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Cancel the common factor.

$\frac{4}{5} + \frac{3 \cdot 2}{5} \cdot \frac{- i}{3} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Rewrite the expression.

$\frac{4}{5} + \frac{2}{5} (- i) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Combine $\frac{2}{5}$ and $i$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Cancel the common factor of $3$ .

Factor $3$ out of $3 i$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{3 (i)}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Cancel the common factor.

$\frac{4}{5} - \frac{2 i}{5} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Rewrite the expression.

$\frac{4}{5} - \frac{2 i}{5} + \frac{i}{5} \cdot 2 + \frac{3 i}{5} (- \frac{i}{3})$

Combine $\frac{i}{5}$ and $2$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{i \cdot 2}{5} + \frac{3 i}{5} (- \frac{i}{3})$

Move $2$ to the left of $i$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{3 i}{5} (- \frac{i}{3})$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{i}{3}$ into the numerator.

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

Factor $3$ out of $3 i$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{3 (i)}{5} \cdot \frac{- i}{3}$

Cancel the common factor.

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

Rewrite the expression.

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{i}{5} (- i)$

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i i}{5}$

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i^{1} i}{5}$

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i^{1} i^{1}}{5}$

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i^{1 + 1}}{5}$

Add $1$ and $1$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i^{2}}{5}$

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{- 1}{5}$

Move the negative in front of the fraction.

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - - \frac{1}{5}$

Multiply $- - \frac{1}{5}$ .

Multiply $- 1$ by $- 1$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + 1 (\frac{1}{5})$

Multiply $\frac{1}{5}$ by $1$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{1}{5}$

Combine the numerators over the common denominator.

$\frac{4 + 1}{5} - \frac{2 i}{5} + \frac{2 i}{5}$

Add $4$ and $1$ .

$\frac{5}{5} - \frac{2 i}{5} + \frac{2 i}{5}$

Add $- \frac{2 i}{5}$ and $\frac{2 i}{5}$ .

$\frac{5}{5} + 0$

Add $\frac{5}{5}$ and $0$ .

$\frac{5}{5}$

Step 4

Divide $5$ by $5$ .

$1$

Simplify (6/5+3/5*i)(2/3-1/3*i)

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Simplify (6/5+3/5i)(2/3-1/3i)