Simplify (6-i)/(6i)+2/i

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

Step 1

Simplify each term.

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

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

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Reorder $6 i$ and $1$ .

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

Simplify the denominator.

Add parentheses.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $6$ by $- 1$ .

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

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

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

Simplify each term.

Move the negative in front of the fraction.

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

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

Factor $6$ out of $6 i$ .

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

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

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

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

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

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

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

Multiply.

Combine.

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

Simplify the denominator.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

Step 2

Subtract $2 i$ from $- i$ .

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

Simplify (6-i)/(6i)+2/i

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