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

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

Step 1

Simplify terms.

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

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

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

$\frac{{\sqrt{3}}^{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{2 i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

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

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

$\frac{{(3^{\frac{1}{2}})}^{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{2 i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

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

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

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

$\frac{3^{\frac{2}{2}}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{2 i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{2 i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Rewrite the expression.

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

Evaluate the exponent.

$\frac{3}{4} + \frac{\sqrt{3}}{2} \cdot \frac{2 i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 i$ .

$\frac{3}{4} + \frac{\sqrt{3}}{2} \cdot \frac{2 (i)}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Cancel the common factor.

$\frac{3}{4} + \frac{\sqrt{3}}{2} \cdot \frac{2 i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Rewrite the expression.

$\frac{3}{4} + \sqrt{3} \frac{i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Combine $\sqrt{3}$ and $\frac{i}{5}$ .

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 i$ .

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{2 (i)}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Cancel the common factor.

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{2 i}{5} \cdot \frac{\sqrt{3}}{2} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Rewrite the expression.

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{i}{5} \sqrt{3} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

Combine $\frac{i}{5}$ and $\sqrt{3}$ .

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5} + \frac{2 i}{5} \cdot \frac{2 i}{5}$

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

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

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5} + \frac{2 i (2 i)}{5 \cdot 5}$

Multiply $2$ by $2$ .

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

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

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5} + \frac{4 (i^{1} i)}{5 \cdot 5}$

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

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5} + \frac{4 (i^{1} i^{1})}{5 \cdot 5}$

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

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5} + \frac{4 i^{1 + 1}}{5 \cdot 5}$

Add $1$ and $1$ .

$\frac{3}{4} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5} + \frac{4 i^{2}}{5 \cdot 5}$

Multiply $5$ by $5$ .

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

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

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

Multiply $4$ by $- 1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Write each expression with a common denominator of $100$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $4$ by $25$ .

$\frac{3 \cdot 25}{100} - \frac{4}{25} \cdot \frac{4}{4} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5}$

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

$\frac{3 \cdot 25}{100} - \frac{4 \cdot 4}{25 \cdot 4} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5}$

Multiply $25$ by $4$ .

$\frac{3 \cdot 25}{100} - \frac{4 \cdot 4}{100} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5}$

Combine the numerators over the common denominator.

$\frac{3 \cdot 25 - 4 \cdot 4}{100} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5}$

Simplify the numerator.

Multiply $3$ by $25$ .

$\frac{75 - 4 \cdot 4}{100} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5}$

Multiply $- 4$ by $4$ .

$\frac{75 - 16}{100} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5}$

Subtract $16$ from $75$ .

$\frac{59}{100} + \frac{\sqrt{3} i}{5} + \frac{i \sqrt{3}}{5}$

Reorder terms.

$\frac{59}{100} + \frac{i \sqrt{3}}{5} + \frac{i \sqrt{3}}{5}$

Combine the numerators over the common denominator.

$\frac{59}{100} + \frac{2 i \sqrt{3}}{5}$

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

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