Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator and denominator of $\frac{1 + i}{i}$ by the conjugate of $i$ to make the denominator real.

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

Step 1.2

Multiply.

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Step 1.2.1

Combine.

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

Step 1.2.2

Simplify the numerator.

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Multiply $i$ by $1$ .

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

Step 1.2.2.3

Multiply $i i$ .

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Step 1.2.2.3.1

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

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

Step 1.2.2.3.2

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

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

Step 1.2.2.3.3

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

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

Step 1.2.2.3.4

Add $1$ and $1$ .

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

Step 1.2.2.4

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

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

Step 1.2.2.5

Reorder $i$ and $- 1$ .

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

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.3.4

Add $1$ and $1$ .

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

Step 1.2.3.5

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

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

Step 1.3

Move the negative one from the denominator of $\frac{- 1 + i}{- 1}$ .

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

Step 1.4

Rewrite $- 1 \cdot (- 1 + i)$ as $- (- 1 + i)$ .

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.7

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

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

Step 1.8

Multiply.

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Step 1.8.1

Combine.

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

Step 1.8.2

Simplify the numerator.

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Step 1.8.2.1

Apply the distributive property.

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

Step 1.8.2.2

Multiply $3$ by $4$ .

$1 - i - \frac{12 + 3 i}{(4 - i) (4 + i)}$

Step 1.8.3

Simplify the denominator.

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Step 1.8.3.1

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

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Step 1.8.3.1.1

Apply the distributive property.

$1 - i - \frac{12 + 3 i}{4 (4 + i) - i (4 + i)}$

Step 1.8.3.1.2

Apply the distributive property.

$1 - i - \frac{12 + 3 i}{4 \cdot 4 + 4 i - i (4 + i)}$

Step 1.8.3.1.3

Apply the distributive property.

$1 - i - \frac{12 + 3 i}{4 \cdot 4 + 4 i - i \cdot 4 - i i}$

Step 1.8.3.2

Simplify.

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Step 1.8.3.2.1

Multiply $4$ by $4$ .

$1 - i - \frac{12 + 3 i}{16 + 4 i - i \cdot 4 - i i}$

Step 1.8.3.2.2

Multiply $4$ by $- 1$ .

$1 - i - \frac{12 + 3 i}{16 + 4 i - 4 i - i i}$

Step 1.8.3.2.3

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

$1 - i - \frac{12 + 3 i}{16 + 4 i - 4 i - (i^{1} i)}$

Step 1.8.3.2.4

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

$1 - i - \frac{12 + 3 i}{16 + 4 i - 4 i - (i^{1} i^{1})}$

Step 1.8.3.2.5

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

$1 - i - \frac{12 + 3 i}{16 + 4 i - 4 i - i^{1 + 1}}$

Step 1.8.3.2.6

Add $1$ and $1$ .

$1 - i - \frac{12 + 3 i}{16 + 4 i - 4 i - i^{2}}$

Step 1.8.3.2.7

Subtract $4 i$ from $4 i$ .

$1 - i - \frac{12 + 3 i}{16 + 0 - i^{2}}$

Step 1.8.3.2.8

Add $16$ and $0$ .

$1 - i - \frac{12 + 3 i}{16 - i^{2}}$

Step 1.8.3.3

Simplify each term.

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Step 1.8.3.3.1

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

$1 - i - \frac{12 + 3 i}{16 - - 1}$

Step 1.8.3.3.2

Multiply $- 1$ by $- 1$ .

$1 - i - \frac{12 + 3 i}{16 + 1}$

Step 1.8.3.4

Add $16$ and $1$ .

$1 - i - \frac{12 + 3 i}{17}$

Step 1.9

Split the fraction $\frac{12 + 3 i}{17}$ into two fractions.

$1 - i - (\frac{12}{17} + \frac{3 i}{17})$

Step 1.10

Apply the distributive property.

$1 - i - \frac{12}{17} - \frac{3 i}{17}$

Step 2

Simplify the expression.

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Step 2.1

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

$\frac{17}{17} - \frac{12}{17} - i - \frac{3 i}{17}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{17 - 12}{17} - i - \frac{3 i}{17}$

Step 2.3

Subtract $12$ from $17$ .

$\frac{5}{17} - i - \frac{3 i}{17}$

Step 3

To write $- i$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$\frac{5}{17} - i \cdot \frac{17}{17} - \frac{3 i}{17}$

Step 4

Combine $- i$ and $\frac{17}{17}$ .

$\frac{5}{17} + \frac{- i \cdot 17}{17} - \frac{3 i}{17}$

Step 5

Simplify the expression.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{5}{17} + \frac{- 20 i}{17}$

Step 5.2

Move the negative in front of the fraction.

$\frac{5}{17} - \frac{20 i}{17}$