Simplify (1/2+( square root of 3)/2*i)^2

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

Step 1

Simplify terms.

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

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

Rewrite ${(\frac{1}{2} + \frac{\sqrt{3} i}{2})}^{2}$ as $(\frac{1}{2} + \frac{\sqrt{3} i}{2}) (\frac{1}{2} + \frac{\sqrt{3} i}{2})$ .

$(\frac{1}{2} + \frac{\sqrt{3} i}{2}) (\frac{1}{2} + \frac{\sqrt{3} i}{2})$

Step 2

Expand $(\frac{1}{2} + \frac{\sqrt{3} i}{2}) (\frac{1}{2} + \frac{\sqrt{3} i}{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{2} (\frac{1}{2} + \frac{\sqrt{3} i}{2}) + \frac{\sqrt{3} i}{2} (\frac{1}{2} + \frac{\sqrt{3} i}{2})$

Apply the distributive property.

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

Apply the distributive property.

$\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{\sqrt{3} i}{2} + \frac{\sqrt{3} i}{2} \cdot \frac{1}{2} + \frac{\sqrt{3} i}{2} \cdot \frac{\sqrt{3} i}{2}$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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

$\frac{1}{2 \cdot 2} + \frac{1}{2} \cdot \frac{\sqrt{3} i}{2} + \frac{\sqrt{3} i}{2} \cdot \frac{1}{2} + \frac{\sqrt{3} i}{2} \cdot \frac{\sqrt{3} i}{2}$

Multiply $2$ by $2$ .

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

Multiply $\frac{1}{2} \cdot \frac{\sqrt{3} i}{2}$ .

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

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

Multiply $2$ by $2$ .

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

Multiply $\frac{\sqrt{3} i}{2} \cdot \frac{1}{2}$ .

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

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

$\frac{1}{4} + \frac{\sqrt{3} i}{4} + \frac{\sqrt{3} i}{4} + \frac{{\sqrt{3}}^{2} i^{2}}{4}$

Simplify the numerator.

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

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

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

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

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

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

$\frac{1}{4} + \frac{\sqrt{3} i}{4} + \frac{\sqrt{3} i}{4} + \frac{3^{\frac{2}{2}} i^{2}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{4} + \frac{\sqrt{3} i}{4} + \frac{\sqrt{3} i}{4} + \frac{3^{\frac{2}{2}} i^{2}}{4}$

Rewrite the expression.

$\frac{1}{4} + \frac{\sqrt{3} i}{4} + \frac{\sqrt{3} i}{4} + \frac{3^{1} i^{2}}{4}$

Evaluate the exponent.

$\frac{1}{4} + \frac{\sqrt{3} i}{4} + \frac{\sqrt{3} i}{4} + \frac{3 i^{2}}{4}$

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

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

Multiply $3$ by $- 1$ .

$\frac{1}{4} + \frac{\sqrt{3} i}{4} + \frac{\sqrt{3} i}{4} + \frac{- 3}{4}$

Move the negative in front of the fraction.

$\frac{1}{4} + \frac{\sqrt{3} i}{4} + \frac{\sqrt{3} i}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$\frac{1 - 3}{4} + \frac{\sqrt{3} i}{4} + \frac{\sqrt{3} i}{4}$

Subtract $3$ from $1$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 \sqrt{3} i$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (1/2+( square root of 3)/2*i)^2

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