Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $i$ and $\frac{5}{4}$ .

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

Step 1.1.2

Move $5$ to the left of $i$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

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

Step 3.1.1.6

Multiply $4$ by $4$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

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

Step 3.1.2.4.2

Rewrite the expression.

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

Step 3.1.2.5

Evaluate the exponent.

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

Step 3.1.3

Cancel the common factor of $2$ and $16$ .

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Step 3.1.3.1

Factor $2$ out of $2$ .

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

Step 3.1.3.2

Cancel the common factors.

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Step 3.1.3.2.1

Factor $2$ out of $16$ .

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

Step 3.1.3.2.2

Cancel the common factor.

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

Step 3.1.3.2.3

Rewrite the expression.

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

Step 3.1.4

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

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Step 3.1.4.1

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

$\frac{1}{8} - \frac{\sqrt{2} (5 i)}{4 \cdot 4} - \frac{5 i}{4} \cdot \frac{\sqrt{2}}{4} - \frac{5 i}{4} (- \frac{5 i}{4})$

Step 3.1.4.2

Multiply $4$ by $4$ .

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

Step 3.1.5

Move $5$ to the left of $\sqrt{2}$ .

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

Multiply $4$ by $4$ .

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

Step 3.1.7

Move $5$ to the left of $\sqrt{2}$ .

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

Step 3.1.8

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

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Step 3.1.8.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.8.2

Multiply $\frac{5 i}{4}$ by $1$ .

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

Step 3.1.8.3

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

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

Step 3.1.8.4

Multiply $5$ by $5$ .

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

Step 3.1.8.5

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

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

Step 3.1.8.6

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

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

Step 3.1.8.7

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

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

Step 3.1.8.8

Add $1$ and $1$ .

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

Step 3.1.8.9

Multiply $4$ by $4$ .

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

Step 3.1.9

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

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

Step 3.1.10

Multiply $25$ by $- 1$ .

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

Step 3.1.11

Move the negative in front of the fraction.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $8$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Subtract $25$ from $2$ .

$\frac{- 23}{16} - \frac{5 \sqrt{2} i}{16} - \frac{5 \sqrt{2} i}{16}$

Step 3.6

Subtract $\frac{5 \sqrt{2} i}{16}$ from $- \frac{5 \sqrt{2} i}{16}$ .

$\frac{- 23}{16} - 2 \frac{5 \sqrt{2} i}{16}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{23}{16} - 2 \frac{5 \sqrt{2} i}{16}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $- 2$ .

$- \frac{23}{16} + 2 (- 1) \frac{5 \sqrt{2} i}{16}$

Step 4.2.2

Factor $2$ out of $16$ .

$- \frac{23}{16} + 2 \cdot - 1 \frac{5 \sqrt{2} i}{2 \cdot 8}$

Step 4.2.3

Cancel the common factor.

$- \frac{23}{16} + 2 \cdot - 1 \frac{5 \sqrt{2} i}{2 \cdot 8}$

Step 4.2.4

Rewrite the expression.

$- \frac{23}{16} - 1 \frac{5 \sqrt{2} i}{8}$

Step 4.3

Rewrite $- 1 \frac{5 \sqrt{2} i}{8}$ as $- \frac{5 \sqrt{2} i}{8}$ .

$- \frac{23}{16} - \frac{5 \sqrt{2} i}{8}$