Simplify (3i-1)/(2i-1)

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

Step 1

Multiply the numerator and denominator of $\frac{3 i - 1}{- 1 + 2 i}$ by the conjugate of $- 1 + 2 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(3 i - 1) (- 1 - 2 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $3$ .

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

Multiply $3 i (- 2 i)$ .

Multiply $- 2$ by $3$ .

$\frac{- 3 i - 6 i i - 1 \cdot - 1 - 1 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

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

$\frac{- 3 i - 6 (i^{1} i) - 1 \cdot - 1 - 1 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

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

$\frac{- 3 i - 6 (i^{1} i^{1}) - 1 \cdot - 1 - 1 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

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

$\frac{- 3 i - 6 i^{1 + 1} - 1 \cdot - 1 - 1 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Add $1$ and $1$ .

$\frac{- 3 i - 6 i^{2} - 1 \cdot - 1 - 1 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

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

$\frac{- 3 i - 6 \cdot - 1 - 1 \cdot - 1 - 1 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Multiply $- 6$ by $- 1$ .

$\frac{- 3 i + 6 - 1 \cdot - 1 - 1 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Multiply $- 1$ by $- 1$ .

$\frac{- 3 i + 6 + 1 - 1 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Multiply $- 2$ by $- 1$ .

$\frac{- 3 i + 6 + 1 + 2 i}{(- 1 + 2 i) (- 1 - 2 i)}$

Add $- 3 i$ and $2 i$ .

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

Add $6$ and $1$ .

$\frac{7 - i}{(- 1 + 2 i) (- 1 - 2 i)}$

Simplify the denominator.

Expand $(- 1 + 2 i) (- 1 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{7 - i}{- 1 (- 1 - 2 i) + 2 i (- 1 - 2 i)}$

Apply the distributive property.

$\frac{7 - i}{- 1 \cdot - 1 - 1 (- 2 i) + 2 i (- 1 - 2 i)}$

Apply the distributive property.

$\frac{7 - i}{- 1 \cdot - 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i)}$

Simplify.

Multiply $- 1$ by $- 1$ .

$\frac{7 - i}{1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i)}$

Multiply $- 2$ by $- 1$ .

$\frac{7 - i}{1 + 2 i + 2 i \cdot - 1 + 2 i (- 2 i)}$

Multiply $- 1$ by $2$ .

$\frac{7 - i}{1 + 2 i - 2 i + 2 i (- 2 i)}$

Multiply $- 2$ by $2$ .

$\frac{7 - i}{1 + 2 i - 2 i - 4 i i}$

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

$\frac{7 - i}{1 + 2 i - 2 i - 4 (i^{1} i)}$

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

$\frac{7 - i}{1 + 2 i - 2 i - 4 (i^{1} i^{1})}$

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

$\frac{7 - i}{1 + 2 i - 2 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{7 - i}{1 + 2 i - 2 i - 4 i^{2}}$

Subtract $2 i$ from $2 i$ .

$\frac{7 - i}{1 + 0 - 4 i^{2}}$

Add $1$ and $0$ .

$\frac{7 - i}{1 - 4 i^{2}}$

Simplify each term.

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

$\frac{7 - i}{1 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{7 - i}{1 + 4}$

Add $1$ and $4$ .

$\frac{7 - i}{5}$

Step 3

Split the fraction $\frac{7 - i}{5}$ into two fractions.

$\frac{7}{5} + \frac{- i}{5}$

Step 4

Move the negative in front of the fraction.

$\frac{7}{5} - \frac{i}{5}$

Simplify (3i-1)/(2i-1)

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