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

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

Step 1

Simplify terms.

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

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

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $3$ .

$\frac{3}{4} + \frac{{\sqrt{3}}^{2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

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

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

$\frac{3}{4} + \frac{{(3^{\frac{1}{2}})}^{2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

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

$\frac{3}{4} + \frac{3^{\frac{1}{2} \cdot 2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

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

$\frac{3}{4} + \frac{3^{\frac{2}{2}} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3}{4} + \frac{3^{\frac{2}{2}} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Rewrite the expression.

$\frac{3}{4} + \frac{3^{1} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Evaluate the exponent.

$\frac{3}{4} + \frac{3 i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3 i$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $3$ by $2$ .

$\frac{3}{4} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

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

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

$\frac{3}{4} + \frac{i}{2} + \frac{{(3^{\frac{1}{2}})}^{2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

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

$\frac{3}{4} + \frac{i}{2} + \frac{3^{\frac{1}{2} \cdot 2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

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

$\frac{3}{4} + \frac{i}{2} + \frac{3^{\frac{2}{2}} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3}{4} + \frac{i}{2} + \frac{3^{\frac{2}{2}} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Rewrite the expression.

$\frac{3}{4} + \frac{i}{2} + \frac{3^{1} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Evaluate the exponent.

$\frac{3}{4} + \frac{i}{2} + \frac{3 i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3 i$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $3$ by $3$ .

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

Simplify the numerator.

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

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

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

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

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

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

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3^{\frac{2}{2}} i^{2}}{9}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3^{\frac{2}{2}} i^{2}}{9}$

Rewrite the expression.

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

Evaluate the exponent.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3 i^{2}}{9}$

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

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

Multiply $3$ by $- 1$ .

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

Cancel the common factor of $- 3$ and $9$ .

Factor $3$ out of $- 3$ .

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

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

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

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

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

Multiply $4$ by $3$ .

$\frac{3 \cdot 3}{12} - \frac{1}{3} \cdot \frac{4}{4} + \frac{i}{2} + \frac{i}{2}$

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\frac{3 \cdot 3}{12} - \frac{4}{3 \cdot 4} + \frac{i}{2} + \frac{i}{2}$

Multiply $3$ by $4$ .

$\frac{3 \cdot 3}{12} - \frac{4}{12} + \frac{i}{2} + \frac{i}{2}$

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 - 4}{12} + \frac{i}{2} + \frac{i}{2}$

Simplify the numerator.

Multiply $3$ by $3$ .

$\frac{9 - 4}{12} + \frac{i}{2} + \frac{i}{2}$

Subtract $4$ from $9$ .

$\frac{5}{12} + \frac{i}{2} + \frac{i}{2}$

Combine the numerators over the common denominator.

$\frac{5}{12} + \frac{2 i}{2}$

Step 4

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5}{12} + \frac{2 i}{2}$

Divide $i$ by $1$ .

$\frac{5}{12} + i$

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

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