Evaluate (3+2i)(-2+3i)^-1*(-i)+1

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

Simplify each term.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Apply the distributive property.

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

Combine $3$ and $\frac{1}{- 2 + 3 i}$ .

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

Multiply $2 i \frac{1}{- 2 + 3 i}$ .

Combine $\frac{1}{- 2 + 3 i}$ and $2$ .

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

Combine $\frac{2}{- 2 + 3 i}$ and $i$ .

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

Combine the numerators over the common denominator.

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

Combine $\frac{3 + 2 i}{- 2 + 3 i}$ and $i$ .

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

Apply the distributive property.

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

Multiply $2 i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

Multiply $2$ by $- 1$ .

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

Reorder $3 i$ and $- 2$ .

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

Cancel the common factor of $- 2 + 3 i$ .

Cancel the common factor.

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

Rewrite the expression.

$- 1 \cdot 1 + 1$

Multiply $- 1$ by $1$ .

$- 1 + 1$

Add $- 1$ and $1$ .

$0$