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

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

Simplify terms.

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

Multiply $5$ by $5$ .

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

Multiply $4$ by $4$ .

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

Multiply $\frac{5}{4} (- \frac{i \sqrt{2}}{2})$ .

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

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

Multiply $4$ by $2$ .

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

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

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

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

Multiply $4$ by $2$ .

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

Multiply $- \frac{i \sqrt{2}}{2} (- \frac{i \sqrt{2}}{2})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{i \sqrt{2}}{2}$ by $1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

Simplify the numerator.

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $2$ .

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

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

$\frac{25}{16} - \frac{5 i \sqrt{2}}{8} - \frac{5 i \sqrt{2}}{8} - \frac{1}{2}$

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

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

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

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

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

Multiply $2$ by $8$ .

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

Combine the numerators over the common denominator.

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

Subtract $8$ from $25$ .

$\frac{17}{16} - \frac{5 i \sqrt{2}}{8} - \frac{5 i \sqrt{2}}{8}$

Subtract $\frac{5 i \sqrt{2}}{8}$ from $- \frac{5 i \sqrt{2}}{8}$ .

$\frac{17}{16} - 2 \frac{5 i \sqrt{2}}{8}$

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$\frac{17}{16} + 2 (- 1) \frac{5 i \sqrt{2}}{8}$

Factor $2$ out of $8$ .

$\frac{17}{16} + 2 \cdot - 1 \frac{5 i \sqrt{2}}{2 \cdot 4}$

Cancel the common factor.

$\frac{17}{16} + 2 \cdot - 1 \frac{5 i \sqrt{2}}{2 \cdot 4}$

Rewrite the expression.

$\frac{17}{16} - 1 \frac{5 i \sqrt{2}}{4}$

Rewrite $- 1 \frac{5 i \sqrt{2}}{4}$ as $- \frac{5 i \sqrt{2}}{4}$ .

$\frac{17}{16} - \frac{5 i \sqrt{2}}{4}$

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

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