Simplify (( square root of 3)/3+5/2i)^2

${(\frac{\sqrt{3}}{3} + \frac{5}{2} i)}^{2}$

Step 1

Simplify terms.

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

${(\frac{\sqrt{3}}{3} + \frac{5 i}{2})}^{2}$

Rewrite ${(\frac{\sqrt{3}}{3} + \frac{5 i}{2})}^{2}$ as $(\frac{\sqrt{3}}{3} + \frac{5 i}{2}) (\frac{\sqrt{3}}{3} + \frac{5 i}{2})$ .

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$\frac{\sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Step 3

Simplify and combine like terms.

Simplify each term.

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

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

$\frac{\sqrt{3} \sqrt{3}}{3 \cdot 3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

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

$\frac{{\sqrt{3}}^{1} \sqrt{3}}{3 \cdot 3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

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

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{3 \cdot 3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

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

$\frac{{\sqrt{3}}^{1 + 1}}{3 \cdot 3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Add $1$ and $1$ .

$\frac{{\sqrt{3}}^{2}}{3 \cdot 3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Multiply $3$ by $3$ .

$\frac{{\sqrt{3}}^{2}}{9} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

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

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

$\frac{{(3^{\frac{1}{2}})}^{2}}{9} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

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

$\frac{3^{\frac{1}{2} \cdot 2}}{9} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

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

$\frac{3^{\frac{2}{2}}}{9} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{9} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Rewrite the expression.

$\frac{3^{1}}{9} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Evaluate the exponent.

$\frac{3}{9} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

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

Factor $3$ out of $3$ .

$\frac{3 (1)}{9} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Cancel the common factors.

Factor $3$ out of $9$ .

$\frac{3 \cdot 1}{3 \cdot 3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Rewrite the expression.

$\frac{1}{3} + \frac{\sqrt{3}}{3} \cdot \frac{5 i}{2} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Multiply $\frac{\sqrt{3}}{3} \cdot \frac{5 i}{2}$ .

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{5 i}{2}$ .

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

Multiply $3$ by $2$ .

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

Move $5$ to the left of $\sqrt{3}$ .

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i}{2} \cdot \frac{\sqrt{3}}{3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Multiply $\frac{5 i}{2} \cdot \frac{\sqrt{3}}{3}$ .

Multiply $\frac{5 i}{2}$ by $\frac{\sqrt{3}}{3}$ .

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{2 \cdot 3} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Multiply $2$ by $3$ .

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{5 i}{2} \cdot \frac{5 i}{2}$

Multiply $\frac{5 i}{2} \cdot \frac{5 i}{2}$ .

Multiply $\frac{5 i}{2}$ by $\frac{5 i}{2}$ .

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

Multiply $5$ by $5$ .

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{25 i i}{2 \cdot 2}$

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

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{25 (i^{1} i)}{2 \cdot 2}$

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

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{25 (i^{1} i^{1})}{2 \cdot 2}$

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

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{25 i^{1 + 1}}{2 \cdot 2}$

Add $1$ and $1$ .

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{25 i^{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{25 i^{2}}{4}$

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

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{25 \cdot - 1}{4}$

Multiply $25$ by $- 1$ .

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} + \frac{- 25}{4}$

Move the negative in front of the fraction.

$\frac{1}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6} - \frac{25}{4}$

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

$\frac{1}{3} \cdot \frac{4}{4} - \frac{25}{4} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

To write $- \frac{25}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} \cdot \frac{4}{4} - \frac{25}{4} \cdot \frac{3}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

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

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

$\frac{4}{3 \cdot 4} - \frac{25}{4} \cdot \frac{3}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

Multiply $3$ by $4$ .

$\frac{4}{12} - \frac{25}{4} \cdot \frac{3}{3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

Multiply $\frac{25}{4}$ by $\frac{3}{3}$ .

$\frac{4}{12} - \frac{25 \cdot 3}{4 \cdot 3} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

Multiply $4$ by $3$ .

$\frac{4}{12} - \frac{25 \cdot 3}{12} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

Combine the numerators over the common denominator.

$\frac{4 - 25 \cdot 3}{12} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

Simplify the numerator.

Multiply $- 25$ by $3$ .

$\frac{4 - 75}{12} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

Subtract $75$ from $4$ .

$\frac{- 71}{12} + \frac{5 \sqrt{3} i}{6} + \frac{5 i \sqrt{3}}{6}$

Reorder terms.

$\frac{- 71}{12} + \frac{5 i \sqrt{3}}{6} + \frac{5 i \sqrt{3}}{6}$

Combine the numerators over the common denominator.

$\frac{- 71}{12} + \frac{10 i \sqrt{3}}{6}$

Step 4

Simplify each term.

Move the negative in front of the fraction.

$- \frac{71}{12} + \frac{10 i \sqrt{3}}{6}$

Cancel the common factor of $10$ and $6$ .

Factor $2$ out of $10 i \sqrt{3}$ .

$- \frac{71}{12} + \frac{2 (5 i \sqrt{3})}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$- \frac{71}{12} + \frac{2 (5 i \sqrt{3})}{2 (3)}$

Cancel the common factor.

$- \frac{71}{12} + \frac{2 (5 i \sqrt{3})}{2 \cdot 3}$

Rewrite the expression.

$- \frac{71}{12} + \frac{5 i \sqrt{3}}{3}$

Simplify (( square root of 3)/3+5/2i)^2

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