Algebra Examples

${(\frac{\sqrt{2}}{3} - \frac{\sqrt{2}}{3} i)}^{2}$

Step 1

Simplify terms.

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Step 1.1

Combine $i$ and $\frac{\sqrt{2}}{3}$ .

${(\frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3})}^{2}$

Step 1.2

Rewrite ${(\frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3})}^{2}$ as $(\frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3}) (\frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3})$ .

$(\frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3}) (\frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3})$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{\sqrt{2}}{3} (\frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} (\frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3})$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $\frac{\sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3}$ .

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Step 3.1.1.1

Multiply $\frac{\sqrt{2}}{3}$ by $\frac{\sqrt{2}}{3}$ .

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

$\frac{{\sqrt{2}}^{2}}{3 \cdot 3} + \frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.1.6

Multiply $3$ by $3$ .

$\frac{{\sqrt{2}}^{2}}{9} + \frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

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 $2^{\frac{1}{2}}$ .

$\frac{{(2^{\frac{1}{2}})}^{2}}{9} + \frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.2.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{9} + \frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.2.3

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

$\frac{2^{\frac{2}{2}}}{9} + \frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

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.

$\frac{2^{\frac{2}{2}}}{9} + \frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.2.4.2

Rewrite the expression.

$\frac{2^{1}}{9} + \frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.2.5

Evaluate the exponent.

$\frac{2}{9} + \frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3}) - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.3

Multiply $\frac{\sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$ .

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Step 3.1.3.1

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

$\frac{2}{9} - \frac{\sqrt{2} (i \sqrt{2})}{3 \cdot 3} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.3.2

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

$\frac{2}{9} - \frac{{\sqrt{2}}^{1} \sqrt{2} i}{3 \cdot 3} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.3.3

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

$\frac{2}{9} - \frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1} i}{3 \cdot 3} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.3.4

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

$\frac{2}{9} - \frac{{\sqrt{2}}^{1 + 1} i}{3 \cdot 3} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.3.5

Add $1$ and $1$ .

$\frac{2}{9} - \frac{{\sqrt{2}}^{2} i}{3 \cdot 3} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.3.6

Multiply $3$ by $3$ .

$\frac{2}{9} - \frac{{\sqrt{2}}^{2} i}{9} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.4

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

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Step 3.1.4.1

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

$\frac{2}{9} - \frac{{(2^{\frac{1}{2}})}^{2} i}{9} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.4.2

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

$\frac{2}{9} - \frac{2^{\frac{1}{2} \cdot 2} i}{9} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.4.3

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

$\frac{2}{9} - \frac{2^{\frac{2}{2}} i}{9} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.4.4.1

Cancel the common factor.

$\frac{2}{9} - \frac{2^{\frac{2}{2}} i}{9} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.4.4.2

Rewrite the expression.

$\frac{2}{9} - \frac{2^{1} i}{9} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.4.5

Evaluate the exponent.

$\frac{2}{9} - \frac{2 i}{9} - \frac{i \sqrt{2}}{3} \cdot \frac{\sqrt{2}}{3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.5

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

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Step 3.1.5.1

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{\sqrt{2} (i \sqrt{2})}{3 \cdot 3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.5.2

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{{\sqrt{2}}^{1} \sqrt{2} i}{3 \cdot 3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.5.3

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1} i}{3 \cdot 3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.5.4

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{{\sqrt{2}}^{1 + 1} i}{3 \cdot 3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.5.5

Add $1$ and $1$ .

$\frac{2}{9} - \frac{2 i}{9} - \frac{{\sqrt{2}}^{2} i}{3 \cdot 3} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.5.6

Multiply $3$ by $3$ .

$\frac{2}{9} - \frac{2 i}{9} - \frac{{\sqrt{2}}^{2} i}{9} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.6

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

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Step 3.1.6.1

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{{(2^{\frac{1}{2}})}^{2} i}{9} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.6.2

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2^{\frac{1}{2} \cdot 2} i}{9} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.6.3

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2^{\frac{2}{2}} i}{9} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

$\frac{2}{9} - \frac{2 i}{9} - \frac{2^{\frac{2}{2}} i}{9} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.6.4.2

Rewrite the expression.

$\frac{2}{9} - \frac{2 i}{9} - \frac{2^{1} i}{9} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.6.5

Evaluate the exponent.

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} - \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$

Step 3.1.7

Multiply $- \frac{i \sqrt{2}}{3} (- \frac{i \sqrt{2}}{3})$ .

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Step 3.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + 1 \frac{i \sqrt{2}}{3} \frac{i \sqrt{2}}{3}$

Step 3.1.7.2

Multiply $\frac{i \sqrt{2}}{3}$ by $1$ .

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i \sqrt{2}}{3} \cdot \frac{i \sqrt{2}}{3}$

Step 3.1.7.3

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i \sqrt{2} (i \sqrt{2})}{3 \cdot 3}$

Step 3.1.7.4

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{1} i \sqrt{2} \sqrt{2}}{3 \cdot 3}$

Step 3.1.7.5

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{1} i^{1} \sqrt{2} \sqrt{2}}{3 \cdot 3}$

Step 3.1.7.6

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{1 + 1} \sqrt{2} \sqrt{2}}{3 \cdot 3}$

Step 3.1.7.7

Add $1$ and $1$ .

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{2} \sqrt{2} \sqrt{2}}{3 \cdot 3}$

Step 3.1.7.8

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{2} ({\sqrt{2}}^{1} \sqrt{2})}{3 \cdot 3}$

Step 3.1.7.9

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{3 \cdot 3}$

Step 3.1.7.10

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{2} {\sqrt{2}}^{1 + 1}}{3 \cdot 3}$

Step 3.1.7.11

Add $1$ and $1$ .

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{2} {\sqrt{2}}^{2}}{3 \cdot 3}$

Step 3.1.7.12

Multiply $3$ by $3$ .

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{i^{2} {\sqrt{2}}^{2}}{9}$

Step 3.1.8

Simplify the numerator.

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Step 3.1.8.1

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{- {\sqrt{2}}^{2}}{9}$

Step 3.1.8.2

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

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Step 3.1.8.2.1

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{- {(2^{\frac{1}{2}})}^{2}}{9}$

Step 3.1.8.2.2

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{- 2^{\frac{1}{2} \cdot 2}}{9}$

Step 3.1.8.2.3

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

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{- 2^{\frac{2}{2}}}{9}$

Step 3.1.8.2.4

Cancel the common factor of $2$ .

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Step 3.1.8.2.4.1

Cancel the common factor.

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{- 2^{\frac{2}{2}}}{9}$

Step 3.1.8.2.4.2

Rewrite the expression.

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{- 2^{1}}{9}$

Step 3.1.8.2.5

Evaluate the exponent.

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{- 1 \cdot 2}{9}$

Step 3.1.9

Multiply $- 1$ by $2$ .

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} + \frac{- 2}{9}$

Step 3.1.10

Move the negative in front of the fraction.

$\frac{2}{9} - \frac{2 i}{9} - \frac{2 i}{9} - \frac{2}{9}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{2 - 2}{9} - \frac{2 i}{9} - \frac{2 i}{9}$

Step 3.3

Subtract $2$ from $2$ .

$\frac{0}{9} - \frac{2 i}{9} - \frac{2 i}{9}$

Step 3.4

Subtract $\frac{2 i}{9}$ from $- \frac{2 i}{9}$ .

$\frac{0}{9} - 2 \frac{2 i}{9}$

Step 4

Simplify each term.

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Step 4.1

Divide $0$ by $9$ .

$0 - 2 \frac{2 i}{9}$

Step 4.2

Multiply $- 2 \frac{2 i}{9}$ .

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Step 4.2.1

Combine $- 2$ and $\frac{2 i}{9}$ .

$0 + \frac{- 2 (2 i)}{9}$

Step 4.2.2

Multiply $2$ by $- 2$ .

$0 + \frac{- 4 i}{9}$

Step 4.3

Move the negative in front of the fraction.

$0 - \frac{4 i}{9}$

Step 5

Subtract $\frac{4 i}{9}$ from $0$ .

$- \frac{4 i}{9}$