Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Multiply.

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Step 1.2.1

Combine.

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

Step 1.2.2

Simplify the numerator.

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Multiply $- i i$ .

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Step 1.2.2.2.1

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

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

Step 1.2.2.2.2

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

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

Step 1.2.2.2.3

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

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

Step 1.2.2.2.4

Add $1$ and $1$ .

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

Step 1.2.2.3

Simplify each term.

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Step 1.2.2.3.1

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

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

Step 1.2.2.3.2

Multiply $- 1$ by $- 1$ .

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

Step 1.2.2.4

Reorder $6 i$ and $1$ .

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

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

Add parentheses.

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.3.4

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

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

Step 1.2.3.5

Add $1$ and $1$ .

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

Step 1.2.3.6

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

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

Step 1.3

Multiply $6$ by $- 1$ .

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

Step 1.4

Split the fraction $\frac{1 + 6 i}{- 6}$ into two fractions.

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

Step 1.5

Simplify each term.

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Step 1.5.1

Move the negative in front of the fraction.

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

Step 1.5.2

Cancel the common factor of $6$ and $- 6$ .

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Step 1.5.2.1

Factor $6$ out of $6 i$ .

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

Step 1.5.2.2

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

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

Step 1.5.3

Rewrite $- 1 \cdot i$ as $- i$ .

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

Step 1.6

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

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

Step 1.7

Multiply.

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Step 1.7.1

Combine.

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

Step 1.7.2

Simplify the denominator.

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Step 1.7.2.1

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

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

Step 1.7.2.2

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

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

Step 1.7.2.3

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

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

Step 1.7.2.4

Add $1$ and $1$ .

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

Step 1.7.2.5

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

Multiply $2$ by $- 1$ .

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

Step 2

Subtract $2 i$ from $- i$ .

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