Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 1.3

Apply the product rule to $\frac{2 + i}{2}$ .

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

Step 1.4

Raise $2$ to the power of $2$ .

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

Step 1.5

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

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

Step 1.6

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

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Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Apply the distributive property.

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

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $2$ by $2$ .

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

Step 1.7.1.2

Move $2$ to the left of $i$ .

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

Step 1.7.1.3

Multiply $i i$ .

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Step 1.7.1.3.1

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

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

Step 1.7.1.3.2

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

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

Step 1.7.1.3.3

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

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

Step 1.7.1.3.4

Add $1$ and $1$ .

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

Step 1.7.1.4

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

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

Step 1.7.2

Subtract $1$ from $4$ .

$\frac{\frac{3 + 2 i + 2 i}{4}}{1 + \frac{i}{2} - 1}$

Step 1.7.3

Add $2 i$ and $2 i$ .

$\frac{\frac{3 + 4 i}{4}}{1 + \frac{i}{2} - 1}$

Step 2

Simplify the denominator.

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Step 2.1

Subtract $1$ from $1$ .

$\frac{\frac{3 + 4 i}{4}}{0 + \frac{i}{2}}$

Step 2.2

Add $0$ and $\frac{i}{2}$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $4$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$\frac{3 + 4 i}{2} \cdot \frac{1}{i}$

Step 5

Multiply $\frac{3 + 4 i}{2}$ by $\frac{1}{i}$ .

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

Step 6

Multiply the numerator and denominator of $\frac{3 + 4 i}{2 i}$ by the conjugate of $2 i$ to make the denominator real.

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

Step 7

Multiply.

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Step 7.1

Combine.

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

Step 7.2

Simplify the numerator.

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Step 7.2.1

Apply the distributive property.

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

Step 7.2.2

Multiply $4 i i$ .

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Step 7.2.2.1

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

$\frac{3 i + 4 (i^{1} i)}{2 i i}$

Step 7.2.2.2

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

$\frac{3 i + 4 (i^{1} i^{1})}{2 i i}$

Step 7.2.2.3

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

$\frac{3 i + 4 i^{1 + 1}}{2 i i}$

Step 7.2.2.4

Add $1$ and $1$ .

$\frac{3 i + 4 i^{2}}{2 i i}$

Step 7.2.3

Simplify each term.

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Step 7.2.3.1

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

$\frac{3 i + 4 \cdot - 1}{2 i i}$

Step 7.2.3.2

Multiply $4$ by $- 1$ .

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

Step 7.2.4

Reorder $3 i$ and $- 4$ .

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

Step 7.3

Simplify the denominator.

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Step 7.3.1

Add parentheses.

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

Step 7.3.2

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

$\frac{- 4 + 3 i}{2 (i^{1} i)}$

Step 7.3.3

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

$\frac{- 4 + 3 i}{2 (i^{1} i^{1})}$

Step 7.3.4

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

$\frac{- 4 + 3 i}{2 i^{1 + 1}}$

Step 7.3.5

Add $1$ and $1$ .

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

Step 7.3.6

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

$\frac{- 4 + 3 i}{2 \cdot - 1}$

Step 8

Multiply $2$ by $- 1$ .

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

Step 9

Split the fraction $\frac{- 4 + 3 i}{- 2}$ into two fractions.

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

Step 10

Simplify each term.

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Step 10.1

Divide $- 4$ by $- 2$ .

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

Step 10.2

Move the negative in front of the fraction.

$2 - \frac{3 i}{2}$