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

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

Simplify terms.

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

Multiply $5$ by $5$ .

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

Multiply $4$ by $4$ .

$\frac{25}{16} + \frac{5}{4} \cdot \frac{\sqrt{5} i}{2} + \frac{\sqrt{5} i}{2} \cdot \frac{5}{4} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

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

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

$\frac{25}{16} + \frac{5 (\sqrt{5} i)}{4 \cdot 2} + \frac{\sqrt{5} i}{2} \cdot \frac{5}{4} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Multiply $4$ by $2$ .

$\frac{25}{16} + \frac{5 (\sqrt{5} i)}{8} + \frac{\sqrt{5} i}{2} \cdot \frac{5}{4} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

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

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{\sqrt{5} i \cdot 5}{2 \cdot 4} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Multiply $2$ by $4$ .

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{\sqrt{5} i \cdot 5}{8} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

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

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{\sqrt{5} i (\sqrt{5} i)}{2 \cdot 2}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{1} \sqrt{5} i i}{2 \cdot 2}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{1} {\sqrt{5}}^{1} i i}{2 \cdot 2}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{1 + 1} i i}{2 \cdot 2}$

Add $1$ and $1$ .

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{2} i i}{2 \cdot 2}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{2} (i^{1} i)}{2 \cdot 2}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{2} (i^{1} i^{1})}{2 \cdot 2}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{2} i^{1 + 1}}{2 \cdot 2}$

Add $1$ and $1$ .

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{2} i^{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{\sqrt{5}}^{2} i^{2}}{4}$

Simplify the numerator.

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

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{{(5^{\frac{1}{2}})}^{2} i^{2}}{4}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{5^{\frac{1}{2} \cdot 2} i^{2}}{4}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{5^{\frac{2}{2}} i^{2}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{5^{\frac{2}{2}} i^{2}}{4}$

Rewrite the expression.

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{5^{1} i^{2}}{4}$

Evaluate the exponent.

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{5 i^{2}}{4}$

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

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{5 \cdot - 1}{4}$

Multiply $5$ by $- 1$ .

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} + \frac{- 5}{4}$

Move the negative in front of the fraction.

$\frac{25}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8} - \frac{5}{4}$

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

$\frac{25}{16} - \frac{5}{4} \cdot \frac{4}{4} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8}$

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

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

$\frac{25}{16} - \frac{5 \cdot 4}{4 \cdot 4} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8}$

Multiply $4$ by $4$ .

$\frac{25}{16} - \frac{5 \cdot 4}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8}$

Combine the numerators over the common denominator.

$\frac{25 - 5 \cdot 4}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8}$

Simplify the numerator.

Multiply $- 5$ by $4$ .

$\frac{25 - 20}{16} + \frac{5 \sqrt{5} i}{8} + \frac{5 \sqrt{5} i}{8}$

Subtract $20$ from $25$ .

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

Combine the numerators over the common denominator.

$\frac{5}{16} + \frac{10 \sqrt{5} i}{8}$

Cancel the common factor of $10$ and $8$ .

Factor $2$ out of $10 \sqrt{5} i$ .

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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