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

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

Step 1

Simplify terms.

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

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

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

Factor $2$ out of $2 i$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

Factor $2$ out of $2 i$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

Simplify the numerator.

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

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

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{{(2^{\frac{1}{2}})}^{2} i^{2}}{4}$

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

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

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

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{2^{\frac{2}{2}} i^{2}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{2^{\frac{2}{2}} i^{2}}{4}$

Rewrite the expression.

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{2^{1} i^{2}}{4}$

Evaluate the exponent.

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{2 i^{2}}{4}$

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

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{2 \cdot - 1}{4}$

Multiply $2$ by $- 1$ .

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{- 2}{4}$

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

Factor $2$ out of $- 2$ .

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{2 (- 1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{2 \cdot - 1}{2 \cdot 2}$

Cancel the common factor.

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{2 \cdot - 1}{2 \cdot 2}$

Rewrite the expression.

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{- 1}{2}$

Move the negative in front of the fraction.

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} - \frac{1}{2}$

Combine the numerators over the common denominator.

$\frac{1 - 1}{2} + \frac{i}{2} + \frac{i}{2}$

Subtract $1$ from $1$ .

$\frac{0}{2} + \frac{i}{2} + \frac{i}{2}$

Combine the numerators over the common denominator.

$\frac{0}{2} + \frac{2 i}{2}$

Step 4

Simplify each term.

Divide $0$ by $2$ .

$0 + \frac{2 i}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$0 + \frac{2 i}{2}$

Divide $i$ by $1$ .

$0 + i$

Step 5

Add $0$ and $i$ .

$i$

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

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