Algebra Examples

$\frac{(\frac{1}{3} - \frac{i}{6})}{(\frac{2}{3} - \frac{i}{3})}$

Step 1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 1.1

Multiply $\frac{\frac{1}{3} - \frac{i}{6}}{\frac{2}{3} - \frac{i}{3}}$ by $\frac{6}{6}$ .

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

Step 1.2

Combine.

$\frac{6 (\frac{1}{3} - \frac{i}{6})}{6 (\frac{2}{3} - \frac{i}{3})}$

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $3$ .

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Step 3.1.1

Factor $3$ out of $6$ .

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

Step 3.1.2

Cancel the common factor.

$\frac{3 \cdot 2 \frac{1}{3} + 6 (- \frac{i}{6})}{6 (\frac{2}{3}) + 6 (- \frac{i}{3})}$

Step 3.1.3

Rewrite the expression.

$\frac{2 + 6 (- \frac{i}{6})}{6 (\frac{2}{3}) + 6 (- \frac{i}{3})}$

Step 3.2

Cancel the common factor of $6$ .

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Step 3.2.1

Move the leading negative in $- \frac{i}{6}$ into the numerator.

$\frac{2 + 6 \frac{- i}{6}}{6 (\frac{2}{3}) + 6 (- \frac{i}{3})}$

Step 3.2.2

Cancel the common factor.

$\frac{2 + 6 \frac{- i}{6}}{6 (\frac{2}{3}) + 6 (- \frac{i}{3})}$

Step 3.2.3

Rewrite the expression.

$\frac{2 - i}{6 (\frac{2}{3}) + 6 (- \frac{i}{3})}$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Factor $3$ out of $6$ .

$\frac{2 - i}{3 (2) \frac{2}{3} + 6 (- \frac{i}{3})}$

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Multiply $2$ by $2$ .

$\frac{2 - i}{4 + 6 (- \frac{i}{3})}$

Step 3.5

Cancel the common factor of $3$ .

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Step 3.5.1

Move the leading negative in $- \frac{i}{3}$ into the numerator.

$\frac{2 - i}{4 + 6 \frac{- i}{3}}$

Step 3.5.2

Factor $3$ out of $6$ .

$\frac{2 - i}{4 + 3 (2) \frac{- i}{3}}$

Step 3.5.3

Cancel the common factor.

$\frac{2 - i}{4 + 3 \cdot 2 \frac{- i}{3}}$

Step 3.5.4

Rewrite the expression.

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

Step 3.6

Multiply $- 1$ by $2$ .

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

Step 4

Multiply the numerator and denominator of $\frac{2 - i}{4 - 2 i}$ by the conjugate of $4 - 2 i$ to make the denominator real.

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

Step 5

Multiply.

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Step 5.1

Combine.

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

Step 5.2

Simplify the numerator.

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Step 5.2.1

Expand $(2 - i) (4 + 2 i)$ using the FOIL Method.

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Step 5.2.1.1

Apply the distributive property.

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Apply the distributive property.

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

Step 5.2.2

Simplify and combine like terms.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $2$ by $4$ .

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

Step 5.2.2.1.2

Multiply $2$ by $2$ .

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

Step 5.2.2.1.3

Multiply $4$ by $- 1$ .

$\frac{8 + 4 i - 4 i - i (2 i)}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.1.4

Multiply $- i (2 i)$ .

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Step 5.2.2.1.4.1

Multiply $2$ by $- 1$ .

$\frac{8 + 4 i - 4 i - 2 i i}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.1.4.2

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

$\frac{8 + 4 i - 4 i - 2 (i^{1} i)}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.1.4.3

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

$\frac{8 + 4 i - 4 i - 2 (i^{1} i^{1})}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.1.4.4

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

$\frac{8 + 4 i - 4 i - 2 i^{1 + 1}}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.1.4.5

Add $1$ and $1$ .

$\frac{8 + 4 i - 4 i - 2 i^{2}}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.1.5

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

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

Step 5.2.2.1.6

Multiply $- 2$ by $- 1$ .

$\frac{8 + 4 i - 4 i + 2}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.2

Add $8$ and $2$ .

$\frac{10 + 4 i - 4 i}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.3

Subtract $4 i$ from $4 i$ .

$\frac{10 + 0}{(4 - 2 i) (4 + 2 i)}$

Step 5.2.2.4

Add $10$ and $0$ .

$\frac{10}{(4 - 2 i) (4 + 2 i)}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Expand $(4 - 2 i) (4 + 2 i)$ using the FOIL Method.

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Step 5.3.1.1

Apply the distributive property.

$\frac{10}{4 (4 + 2 i) - 2 i (4 + 2 i)}$

Step 5.3.1.2

Apply the distributive property.

$\frac{10}{4 \cdot 4 + 4 (2 i) - 2 i (4 + 2 i)}$

Step 5.3.1.3

Apply the distributive property.

$\frac{10}{4 \cdot 4 + 4 (2 i) - 2 i \cdot 4 - 2 i (2 i)}$

Step 5.3.2

Simplify.

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Step 5.3.2.1

Multiply $4$ by $4$ .

$\frac{10}{16 + 4 (2 i) - 2 i \cdot 4 - 2 i (2 i)}$

Step 5.3.2.2

Multiply $2$ by $4$ .

$\frac{10}{16 + 8 i - 2 i \cdot 4 - 2 i (2 i)}$

Step 5.3.2.3

Multiply $4$ by $- 2$ .

$\frac{10}{16 + 8 i - 8 i - 2 i (2 i)}$

Step 5.3.2.4

Multiply $2$ by $- 2$ .

$\frac{10}{16 + 8 i - 8 i - 4 i i}$

Step 5.3.2.5

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

$\frac{10}{16 + 8 i - 8 i - 4 (i^{1} i)}$

Step 5.3.2.6

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

$\frac{10}{16 + 8 i - 8 i - 4 (i^{1} i^{1})}$

Step 5.3.2.7

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

$\frac{10}{16 + 8 i - 8 i - 4 i^{1 + 1}}$

Step 5.3.2.8

Add $1$ and $1$ .

$\frac{10}{16 + 8 i - 8 i - 4 i^{2}}$

Step 5.3.2.9

Subtract $8 i$ from $8 i$ .

$\frac{10}{16 + 0 - 4 i^{2}}$

Step 5.3.2.10

Add $16$ and $0$ .

$\frac{10}{16 - 4 i^{2}}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

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

$\frac{10}{16 - 4 \cdot - 1}$

Step 5.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{10}{16 + 4}$

Step 5.3.4

Add $16$ and $4$ .

$\frac{10}{20}$

Step 6

Cancel the common factor of $10$ and $20$ .

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Step 6.1

Factor $10$ out of $10$ .

$\frac{10 (1)}{20}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $10$ out of $20$ .

$\frac{10 \cdot 1}{10 \cdot 2}$

Step 6.2.2

Cancel the common factor.

$\frac{10 \cdot 1}{10 \cdot 2}$

Step 6.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$