Algebra Examples

${(\frac{2}{3} - \frac{3}{2} i)}^{2}$

Step 1

Simplify terms.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Combine $i$ and $\frac{3}{2}$ .

${(\frac{2}{3} - \frac{i \cdot 3}{2})}^{2}$

Step 1.1.2

Move $3$ to the left of $i$ .

${(\frac{2}{3} - \frac{3 i}{2})}^{2}$

Step 1.2

Rewrite ${(\frac{2}{3} - \frac{3 i}{2})}^{2}$ as $(\frac{2}{3} - \frac{3 i}{2}) (\frac{2}{3} - \frac{3 i}{2})$ .

$(\frac{2}{3} - \frac{3 i}{2}) (\frac{2}{3} - \frac{3 i}{2})$

Step 2

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

Tap for more steps...

Step 2.1

Apply the distributive property.

$\frac{2}{3} (\frac{2}{3} - \frac{3 i}{2}) - \frac{3 i}{2} (\frac{2}{3} - \frac{3 i}{2})$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Multiply $\frac{2}{3} \cdot \frac{2}{3}$ .

Tap for more steps...

Step 3.1.1.1

Multiply $\frac{2}{3}$ by $\frac{2}{3}$ .

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

Step 3.1.1.2

Multiply $2$ by $2$ .

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

Step 3.1.1.3

Multiply $3$ by $3$ .

$\frac{4}{9} + \frac{2}{3} (- \frac{3 i}{2}) - \frac{3 i}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.2.1

Move the leading negative in $- \frac{3 i}{2}$ into the numerator.

$\frac{4}{9} + \frac{2}{3} \cdot \frac{- 3 i}{2} - \frac{3 i}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2.2

Cancel the common factor.

$\frac{4}{9} + \frac{2}{3} \cdot \frac{- 3 i}{2} - \frac{3 i}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2.3

Rewrite the expression.

$\frac{4}{9} + \frac{1}{3} (- 3 i) - \frac{3 i}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.1.3.1

Factor $3$ out of $- 3 i$ .

$\frac{4}{9} + \frac{1}{3} (3 (- i)) - \frac{3 i}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.3.2

Cancel the common factor.

$\frac{4}{9} + \frac{1}{3} (3 (- i)) - \frac{3 i}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.3.3

Rewrite the expression.

$\frac{4}{9} - i - \frac{3 i}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.1.4.1

Move the leading negative in $- \frac{3 i}{2}$ into the numerator.

$\frac{4}{9} - i + \frac{- 3 i}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.4.2

Factor $3$ out of $- 3 i$ .

$\frac{4}{9} - i + \frac{3 (- i)}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.4.3

Cancel the common factor.

$\frac{4}{9} - i + \frac{3 (- i)}{2} \cdot \frac{2}{3} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.4.4

Rewrite the expression.

$\frac{4}{9} - i + \frac{- i}{2} \cdot 2 - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.5.1

Cancel the common factor.

$\frac{4}{9} - i + \frac{- i}{2} \cdot 2 - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.5.2

Rewrite the expression.

$\frac{4}{9} - i - i - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.6

Multiply $- \frac{3 i}{2} (- \frac{3 i}{2})$ .

Tap for more steps...

Step 3.1.6.1

Multiply $- 1$ by $- 1$ .

$\frac{4}{9} - i - i + 1 \frac{3 i}{2} \frac{3 i}{2}$

Step 3.1.6.2

Multiply $\frac{3 i}{2}$ by $1$ .

$\frac{4}{9} - i - i + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.6.3

Multiply $\frac{3 i}{2}$ by $\frac{3 i}{2}$ .

$\frac{4}{9} - i - i + \frac{3 i (3 i)}{2 \cdot 2}$

Step 3.1.6.4

Multiply $3$ by $3$ .

$\frac{4}{9} - i - i + \frac{9 i i}{2 \cdot 2}$

Step 3.1.6.5

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

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

Step 3.1.6.6

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

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

Step 3.1.6.7

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

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

Step 3.1.6.8

Add $1$ and $1$ .

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

Step 3.1.6.9

Multiply $2$ by $2$ .

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

Step 3.1.7

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

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

Step 3.1.8

Multiply $9$ by $- 1$ .

$\frac{4}{9} - i - i + \frac{- 9}{4}$

Step 3.1.9

Move the negative in front of the fraction.

$\frac{4}{9} - i - i - \frac{9}{4}$

Step 3.2

To write $\frac{4}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4}{9} \cdot \frac{4}{4} - \frac{9}{4} - i - i$

Step 3.3

To write $- \frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{4}{9} \cdot \frac{4}{4} - \frac{9}{4} \cdot \frac{9}{9} - i - i$

Step 3.4

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.4.1

Multiply $\frac{4}{9}$ by $\frac{4}{4}$ .

$\frac{4 \cdot 4}{9 \cdot 4} - \frac{9}{4} \cdot \frac{9}{9} - i - i$

Step 3.4.2

Multiply $9$ by $4$ .

$\frac{4 \cdot 4}{36} - \frac{9}{4} \cdot \frac{9}{9} - i - i$

Step 3.4.3

Multiply $\frac{9}{4}$ by $\frac{9}{9}$ .

$\frac{4 \cdot 4}{36} - \frac{9 \cdot 9}{4 \cdot 9} - i - i$

Step 3.4.4

Multiply $4$ by $9$ .

$\frac{4 \cdot 4}{36} - \frac{9 \cdot 9}{36} - i - i$

Step 3.5

Combine the numerators over the common denominator.

$\frac{4 \cdot 4 - 9 \cdot 9}{36} - i - i$

Step 3.6

Simplify the numerator.

Tap for more steps...

Step 3.6.1

Multiply $4$ by $4$ .

$\frac{16 - 9 \cdot 9}{36} - i - i$

Step 3.6.2

Multiply $- 9$ by $9$ .

$\frac{16 - 81}{36} - i - i$

Step 3.6.3

Subtract $81$ from $16$ .

$\frac{- 65}{36} - i - i$

Step 3.7

Subtract $i$ from $- i$ .

$\frac{- 65}{36} - 2 i$

Step 4

Move the negative in front of the fraction.

$- \frac{65}{36} - 2 i$