Simplify (( square root of 5)/2+3/4i)^2

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

Step 1

Simplify terms.

Combine $\frac{3}{4}$ and $i$ .

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

Rewrite ${(\frac{\sqrt{5}}{2} + \frac{3 i}{4})}^{2}$ as $(\frac{\sqrt{5}}{2} + \frac{3 i}{4}) (\frac{\sqrt{5}}{2} + \frac{3 i}{4})$ .

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

Step 2

Expand $(\frac{\sqrt{5}}{2} + \frac{3 i}{4}) (\frac{\sqrt{5}}{2} + \frac{3 i}{4})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

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

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

Raise $\sqrt{5}$ to the power of $1$ .

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

Raise $\sqrt{5}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

Multiply $\frac{\sqrt{5}}{2}$ by $\frac{3 i}{4}$ .

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

Multiply $2$ by $4$ .

$\frac{5}{4} + \frac{\sqrt{5} (3 i)}{8} + \frac{3 i}{4} \cdot \frac{\sqrt{5}}{2} + \frac{3 i}{4} \cdot \frac{3 i}{4}$

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

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i}{4} \cdot \frac{\sqrt{5}}{2} + \frac{3 i}{4} \cdot \frac{3 i}{4}$

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

Multiply $\frac{3 i}{4}$ by $\frac{\sqrt{5}}{2}$ .

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{4 \cdot 2} + \frac{3 i}{4} \cdot \frac{3 i}{4}$

Multiply $4$ by $2$ .

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{3 i}{4} \cdot \frac{3 i}{4}$

Multiply $\frac{3 i}{4} \cdot \frac{3 i}{4}$ .

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

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{3 i (3 i)}{4 \cdot 4}$

Multiply $3$ by $3$ .

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{9 i i}{4 \cdot 4}$

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

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{9 (i^{1} i)}{4 \cdot 4}$

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

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{9 (i^{1} i^{1})}{4 \cdot 4}$

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

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{9 i^{1 + 1}}{4 \cdot 4}$

Add $1$ and $1$ .

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{9 i^{2}}{4 \cdot 4}$

Multiply $4$ by $4$ .

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{9 i^{2}}{16}$

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

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{9 \cdot - 1}{16}$

Multiply $9$ by $- 1$ .

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} + \frac{- 9}{16}$

Move the negative in front of the fraction.

$\frac{5}{4} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8} - \frac{9}{16}$

To write $\frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{5}{4} \cdot \frac{4}{4} - \frac{9}{16} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8}$

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

Multiply $\frac{5}{4}$ by $\frac{4}{4}$ .

$\frac{5 \cdot 4}{4 \cdot 4} - \frac{9}{16} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8}$

Multiply $4$ by $4$ .

$\frac{5 \cdot 4}{16} - \frac{9}{16} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8}$

Combine the numerators over the common denominator.

$\frac{5 \cdot 4 - 9}{16} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8}$

Simplify the numerator.

Multiply $5$ by $4$ .

$\frac{20 - 9}{16} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8}$

Subtract $9$ from $20$ .

$\frac{11}{16} + \frac{3 \sqrt{5} i}{8} + \frac{3 i \sqrt{5}}{8}$

Reorder terms.

$\frac{11}{16} + \frac{3 i \sqrt{5}}{8} + \frac{3 i \sqrt{5}}{8}$

Combine the numerators over the common denominator.

$\frac{11}{16} + \frac{6 i \sqrt{5}}{8}$

Step 4

Cancel the common factor of $6$ and $8$ .

Factor $2$ out of $6 i \sqrt{5}$ .

$\frac{11}{16} + \frac{2 (3 i \sqrt{5})}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{11}{16} + \frac{2 (3 i \sqrt{5})}{2 (4)}$

Cancel the common factor.

$\frac{11}{16} + \frac{2 (3 i \sqrt{5})}{2 \cdot 4}$

Rewrite the expression.

$\frac{11}{16} + \frac{3 i \sqrt{5}}{4}$

Simplify (( square root of 5)/2+3/4i)^2

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