Algebra Examples

${(\frac{\sqrt{3}}{5} - \frac{\sqrt{2}}{5} i)}^{2}$

Step 1

Simplify terms.

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Step 1.1

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

${(\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5})}^{2}$

Step 1.2

Rewrite ${(\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5})}^{2}$ as $(\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5}) (\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5})$ .

$(\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5}) (\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5})$

Step 2

Expand $(\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5}) (\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{\sqrt{3}}{5} (\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} (\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5})$

Step 2.2

Apply the distributive property.

$\frac{\sqrt{3}}{5} \cdot \frac{\sqrt{3}}{5} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} (\frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5})$

Step 2.3

Apply the distributive property.

$\frac{\sqrt{3}}{5} \cdot \frac{\sqrt{3}}{5} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

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 $\frac{\sqrt{3}}{5} \cdot \frac{\sqrt{3}}{5}$ .

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Step 3.1.1.1

Multiply $\frac{\sqrt{3}}{5}$ by $\frac{\sqrt{3}}{5}$ .

$\frac{\sqrt{3} \sqrt{3}}{5 \cdot 5} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} \sqrt{3}}{5 \cdot 5} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.1.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{5 \cdot 5} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.1.4

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

$\frac{{\sqrt{3}}^{1 + 1}}{5 \cdot 5} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.1.5

Add $1$ and $1$ .

$\frac{{\sqrt{3}}^{2}}{5 \cdot 5} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.1.6

Multiply $5$ by $5$ .

$\frac{{\sqrt{3}}^{2}}{25} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{25} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.2.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{25} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.2.3

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

$\frac{3^{\frac{2}{2}}}{25} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

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.

$\frac{3^{\frac{2}{2}}}{25} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.2.4.2

Rewrite the expression.

$\frac{3^{1}}{25} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.2.5

Evaluate the exponent.

$\frac{3}{25} + \frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.3

Multiply $\frac{\sqrt{3}}{5} (- \frac{i \sqrt{2}}{5})$ .

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Step 3.1.3.1

Multiply $\frac{\sqrt{3}}{5}$ by $\frac{i \sqrt{2}}{5}$ .

$\frac{3}{25} - \frac{\sqrt{3} (i \sqrt{2})}{5 \cdot 5} - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.3.2

Combine using the product rule for radicals.

$\frac{3}{25} - \frac{\sqrt{2 \cdot 3} i}{5 \cdot 5} - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.3.3

Multiply $2$ by $3$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{5 \cdot 5} - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.3.4

Multiply $5$ by $5$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.4

Multiply $- \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{3}}{5}$ .

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Step 3.1.4.1

Multiply $\frac{\sqrt{3}}{5}$ by $\frac{i \sqrt{2}}{5}$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{3} (i \sqrt{2})}{5 \cdot 5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.4.2

Combine using the product rule for radicals.

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{2 \cdot 3} i}{5 \cdot 5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.4.3

Multiply $2$ by $3$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{5 \cdot 5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.4.4

Multiply $5$ by $5$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Step 3.1.5

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

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + 1 \frac{i \sqrt{2}}{5} \frac{i \sqrt{2}}{5}$

Step 3.1.5.2

Multiply $\frac{i \sqrt{2}}{5}$ by $1$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i \sqrt{2}}{5} \cdot \frac{i \sqrt{2}}{5}$

Step 3.1.5.3

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i \sqrt{2} (i \sqrt{2})}{5 \cdot 5}$

Step 3.1.5.4

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{1} i \sqrt{2} \sqrt{2}}{5 \cdot 5}$

Step 3.1.5.5

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{1} i^{1} \sqrt{2} \sqrt{2}}{5 \cdot 5}$

Step 3.1.5.6

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{1 + 1} \sqrt{2} \sqrt{2}}{5 \cdot 5}$

Step 3.1.5.7

Add $1$ and $1$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{2} \sqrt{2} \sqrt{2}}{5 \cdot 5}$

Step 3.1.5.8

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{2} ({\sqrt{2}}^{1} \sqrt{2})}{5 \cdot 5}$

Step 3.1.5.9

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{5 \cdot 5}$

Step 3.1.5.10

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{2} {\sqrt{2}}^{1 + 1}}{5 \cdot 5}$

Step 3.1.5.11

Add $1$ and $1$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{2} {\sqrt{2}}^{2}}{5 \cdot 5}$

Step 3.1.5.12

Multiply $5$ by $5$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{i^{2} {\sqrt{2}}^{2}}{25}$

Step 3.1.6

Simplify the numerator.

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Step 3.1.6.1

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{- {\sqrt{2}}^{2}}{25}$

Step 3.1.6.2

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

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Step 3.1.6.2.1

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{- {(2^{\frac{1}{2}})}^{2}}{25}$

Step 3.1.6.2.2

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{- 2^{\frac{1}{2} \cdot 2}}{25}$

Step 3.1.6.2.3

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

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{- 2^{\frac{2}{2}}}{25}$

Step 3.1.6.2.4

Cancel the common factor of $2$ .

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Step 3.1.6.2.4.1

Cancel the common factor.

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{- 2^{\frac{2}{2}}}{25}$

Step 3.1.6.2.4.2

Rewrite the expression.

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{- 2^{1}}{25}$

Step 3.1.6.2.5

Evaluate the exponent.

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{- 1 \cdot 2}{25}$

Step 3.1.7

Multiply $- 1$ by $2$ .

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} + \frac{- 2}{25}$

Step 3.1.8

Move the negative in front of the fraction.

$\frac{3}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25} - \frac{2}{25}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 - 2}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25}$

Step 3.3

Subtract $2$ from $3$ .

$\frac{1}{25} - \frac{\sqrt{6} i}{25} - \frac{\sqrt{6} i}{25}$

Step 3.4

Subtract $\frac{\sqrt{6} i}{25}$ from $- \frac{\sqrt{6} i}{25}$ .

$\frac{1}{25} - 2 \frac{\sqrt{6} i}{25}$

Step 4

Simplify each term.

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Step 4.1

Combine $- 2$ and $\frac{\sqrt{6} i}{25}$ .

$\frac{1}{25} + \frac{- 2 (\sqrt{6} i)}{25}$

Step 4.2

Move the negative in front of the fraction.

$\frac{1}{25} - \frac{2 \sqrt{6} i}{25}$