Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Multiply.

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Step 1.2.1

Combine.

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

Step 1.2.2

Simplify the numerator.

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Multiply $i i$ .

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Step 1.2.2.2.1

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

$\frac{2 i + i^{1} i}{i i} - \frac{4}{6 - i}$

Step 1.2.2.2.2

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

$\frac{2 i + i^{1} i^{1}}{i i} - \frac{4}{6 - i}$

Step 1.2.2.2.3

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

$\frac{2 i + i^{1 + 1}}{i i} - \frac{4}{6 - i}$

Step 1.2.2.2.4

Add $1$ and $1$ .

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

Step 1.2.2.3

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

$\frac{2 i - 1}{i i} - \frac{4}{6 - i}$

Step 1.2.2.4

Reorder $2 i$ and $- 1$ .

$\frac{- 1 + 2 i}{i i} - \frac{4}{6 - i}$

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

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

$\frac{- 1 + 2 i}{i^{1} i} - \frac{4}{6 - i}$

Step 1.2.3.2

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

$\frac{- 1 + 2 i}{i^{1} i^{1}} - \frac{4}{6 - i}$

Step 1.2.3.3

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

$\frac{- 1 + 2 i}{i^{1 + 1}} - \frac{4}{6 - i}$

Step 1.2.3.4

Add $1$ and $1$ .

$\frac{- 1 + 2 i}{i^{2}} - \frac{4}{6 - i}$

Step 1.2.3.5

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

$\frac{- 1 + 2 i}{- 1} - \frac{4}{6 - i}$

Step 1.3

Move the negative one from the denominator of $\frac{- 1 + 2 i}{- 1}$ .

$- 1 \cdot (- 1 + 2 i) - \frac{4}{6 - i}$

Step 1.4

Rewrite $- 1 \cdot (- 1 + 2 i)$ as $- (- 1 + 2 i)$ .

$- (- 1 + 2 i) - \frac{4}{6 - i}$

Step 1.5

Apply the distributive property.

$- - 1 - (2 i) - \frac{4}{6 - i}$

Step 1.6

Multiply $- 1$ by $- 1$ .

$1 - (2 i) - \frac{4}{6 - i}$

Step 1.7

Multiply $2$ by $- 1$ .

$1 - 2 i - \frac{4}{6 - i}$

Step 1.8

Multiply the numerator and denominator of $\frac{4}{6 - i}$ by the conjugate of $6 - i$ to make the denominator real.

$1 - 2 i - (\frac{4}{6 - i} \cdot \frac{6 + i}{6 + i})$

Step 1.9

Multiply.

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Step 1.9.1

Combine.

$1 - 2 i - \frac{4 (6 + i)}{(6 - i) (6 + i)}$

Step 1.9.2

Simplify the numerator.

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Step 1.9.2.1

Apply the distributive property.

$1 - 2 i - \frac{4 \cdot 6 + 4 i}{(6 - i) (6 + i)}$

Step 1.9.2.2

Multiply $4$ by $6$ .

$1 - 2 i - \frac{24 + 4 i}{(6 - i) (6 + i)}$

Step 1.9.3

Simplify the denominator.

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Step 1.9.3.1

Expand $(6 - i) (6 + i)$ using the FOIL Method.

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Step 1.9.3.1.1

Apply the distributive property.

$1 - 2 i - \frac{24 + 4 i}{6 (6 + i) - i (6 + i)}$

Step 1.9.3.1.2

Apply the distributive property.

$1 - 2 i - \frac{24 + 4 i}{6 \cdot 6 + 6 i - i (6 + i)}$

Step 1.9.3.1.3

Apply the distributive property.

$1 - 2 i - \frac{24 + 4 i}{6 \cdot 6 + 6 i - i \cdot 6 - i i}$

Step 1.9.3.2

Simplify.

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Step 1.9.3.2.1

Multiply $6$ by $6$ .

$1 - 2 i - \frac{24 + 4 i}{36 + 6 i - i \cdot 6 - i i}$

Step 1.9.3.2.2

Multiply $6$ by $- 1$ .

$1 - 2 i - \frac{24 + 4 i}{36 + 6 i - 6 i - i i}$

Step 1.9.3.2.3

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

$1 - 2 i - \frac{24 + 4 i}{36 + 6 i - 6 i - (i^{1} i)}$

Step 1.9.3.2.4

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

$1 - 2 i - \frac{24 + 4 i}{36 + 6 i - 6 i - (i^{1} i^{1})}$

Step 1.9.3.2.5

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

$1 - 2 i - \frac{24 + 4 i}{36 + 6 i - 6 i - i^{1 + 1}}$

Step 1.9.3.2.6

Add $1$ and $1$ .

$1 - 2 i - \frac{24 + 4 i}{36 + 6 i - 6 i - i^{2}}$

Step 1.9.3.2.7

Subtract $6 i$ from $6 i$ .

$1 - 2 i - \frac{24 + 4 i}{36 + 0 - i^{2}}$

Step 1.9.3.2.8

Add $36$ and $0$ .

$1 - 2 i - \frac{24 + 4 i}{36 - i^{2}}$

Step 1.9.3.3

Simplify each term.

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Step 1.9.3.3.1

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

$1 - 2 i - \frac{24 + 4 i}{36 - - 1}$

Step 1.9.3.3.2

Multiply $- 1$ by $- 1$ .

$1 - 2 i - \frac{24 + 4 i}{36 + 1}$

Step 1.9.3.4

Add $36$ and $1$ .

$1 - 2 i - \frac{24 + 4 i}{37}$

Step 1.10

Split the fraction $\frac{24 + 4 i}{37}$ into two fractions.

$1 - 2 i - (\frac{24}{37} + \frac{4 i}{37})$

Step 1.11

Apply the distributive property.

$1 - 2 i - \frac{24}{37} - \frac{4 i}{37}$

Step 2

Simplify the expression.

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Step 2.1

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

$\frac{37}{37} - \frac{24}{37} - 2 i - \frac{4 i}{37}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{37 - 24}{37} - 2 i - \frac{4 i}{37}$

Step 2.3

Subtract $24$ from $37$ .

$\frac{13}{37} - 2 i - \frac{4 i}{37}$

Step 3

To write $- 2 i$ as a fraction with a common denominator, multiply by $\frac{37}{37}$ .

$\frac{13}{37} - 2 i \cdot \frac{37}{37} - \frac{4 i}{37}$

Step 4

Combine $- 2 i$ and $\frac{37}{37}$ .

$\frac{13}{37} + \frac{- 2 i \cdot 37}{37} - \frac{4 i}{37}$

Step 5

Simplify the expression.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{13}{37} + \frac{- 78 i}{37}$

Step 5.2

Move the negative in front of the fraction.

$\frac{13}{37} - \frac{78 i}{37}$