Algebra Examples

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

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 $\frac{3}{2}$ .

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

Step 1.1.2

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

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

Step 1.2

Rewrite ${(\frac{\sqrt{3}}{4} - \frac{3 i}{2})}^{2}$ as $(\frac{\sqrt{3}}{4} - \frac{3 i}{2}) (\frac{\sqrt{3}}{4} - \frac{3 i}{2})$ .

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

Step 2

Expand $(\frac{\sqrt{3}}{4} - \frac{3 i}{2}) (\frac{\sqrt{3}}{4} - \frac{3 i}{2})$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $\frac{\sqrt{3}}{4} \cdot \frac{\sqrt{3}}{4}$ .

Tap for more steps...

Step 3.1.1.1

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

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

Step 3.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 3.1.1.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

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

Step 3.1.1.6

Multiply $4$ by $4$ .

$\frac{{\sqrt{3}}^{2}}{16} + \frac{\sqrt{3}}{4} (- \frac{3 i}{2}) - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{16} + \frac{\sqrt{3}}{4} (- \frac{3 i}{2}) - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3^{\frac{1}{2} \cdot 2}}{16} + \frac{\sqrt{3}}{4} (- \frac{3 i}{2}) - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3^{\frac{2}{2}}}{16} + \frac{\sqrt{3}}{4} (- \frac{3 i}{2}) - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.2.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{16} + \frac{\sqrt{3}}{4} (- \frac{3 i}{2}) - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2.4.2

Rewrite the expression.

$\frac{3^{1}}{16} + \frac{\sqrt{3}}{4} (- \frac{3 i}{2}) - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.2.5

Evaluate the exponent.

$\frac{3}{16} + \frac{\sqrt{3}}{4} (- \frac{3 i}{2}) - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.3

Multiply $\frac{\sqrt{3}}{4} (- \frac{3 i}{2})$ .

Tap for more steps...

Step 3.1.3.1

Multiply $\frac{\sqrt{3}}{4}$ by $\frac{3 i}{2}$ .

$\frac{3}{16} - \frac{\sqrt{3} (3 i)}{4 \cdot 2} - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.3.2

Multiply $4$ by $2$ .

$\frac{3}{16} - \frac{\sqrt{3} (3 i)}{8} - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.4

Move $3$ to the left of $\sqrt{3}$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.5

Multiply $- \frac{3 i}{2} \cdot \frac{\sqrt{3}}{4}$ .

Tap for more steps...

Step 3.1.5.1

Multiply $\frac{\sqrt{3}}{4}$ by $\frac{3 i}{2}$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{\sqrt{3} (3 i)}{4 \cdot 2} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.5.2

Multiply $4$ by $2$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{\sqrt{3} (3 i)}{8} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.6

Move $3$ to the left of $\sqrt{3}$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} - \frac{3 i}{2} (- \frac{3 i}{2})$

Step 3.1.7

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

Tap for more steps...

Step 3.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + 1 \frac{3 i}{2} \frac{3 i}{2}$

Step 3.1.7.2

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

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.7.3

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

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{3 i (3 i)}{2 \cdot 2}$

Step 3.1.7.4

Multiply $3$ by $3$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{9 i i}{2 \cdot 2}$

Step 3.1.7.5

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

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{9 (i^{1} i)}{2 \cdot 2}$

Step 3.1.7.6

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

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{9 (i^{1} i^{1})}{2 \cdot 2}$

Step 3.1.7.7

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

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{9 i^{1 + 1}}{2 \cdot 2}$

Step 3.1.7.8

Add $1$ and $1$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{9 i^{2}}{2 \cdot 2}$

Step 3.1.7.9

Multiply $2$ by $2$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{9 i^{2}}{4}$

Step 3.1.8

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

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{9 \cdot - 1}{4}$

Step 3.1.9

Multiply $9$ by $- 1$ .

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} + \frac{- 9}{4}$

Step 3.1.10

Move the negative in front of the fraction.

$\frac{3}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8} - \frac{9}{4}$

Step 3.2

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

$\frac{3}{16} - \frac{9}{4} \cdot \frac{4}{4} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8}$

Step 3.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.3.1

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

$\frac{3}{16} - \frac{9 \cdot 4}{4 \cdot 4} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8}$

Step 3.3.2

Multiply $4$ by $4$ .

$\frac{3}{16} - \frac{9 \cdot 4}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{3 - 9 \cdot 4}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8}$

Step 3.5

Simplify the numerator.

Tap for more steps...

Step 3.5.1

Multiply $- 9$ by $4$ .

$\frac{3 - 36}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8}$

Step 3.5.2

Subtract $36$ from $3$ .

$\frac{- 33}{16} - \frac{3 \sqrt{3} i}{8} - \frac{3 \sqrt{3} i}{8}$

Step 3.6

Subtract $\frac{3 \sqrt{3} i}{8}$ from $- \frac{3 \sqrt{3} i}{8}$ .

$\frac{- 33}{16} - 2 \frac{3 \sqrt{3} i}{8}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Move the negative in front of the fraction.

$- \frac{33}{16} - 2 \frac{3 \sqrt{3} i}{8}$

Step 4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1

Factor $2$ out of $- 2$ .

$- \frac{33}{16} + 2 (- 1) \frac{3 \sqrt{3} i}{8}$

Step 4.2.2

Factor $2$ out of $8$ .

$- \frac{33}{16} + 2 \cdot - 1 \frac{3 \sqrt{3} i}{2 \cdot 4}$

Step 4.2.3

Cancel the common factor.

$- \frac{33}{16} + 2 \cdot - 1 \frac{3 \sqrt{3} i}{2 \cdot 4}$

Step 4.2.4

Rewrite the expression.

$- \frac{33}{16} - 1 \frac{3 \sqrt{3} i}{4}$

Step 4.3

Rewrite $- 1 \frac{3 \sqrt{3} i}{4}$ as $- \frac{3 \sqrt{3} i}{4}$ .

$- \frac{33}{16} - \frac{3 \sqrt{3} i}{4}$