Question

algebra 2 / trig
multiplying and dividing complex numbers
date
© 2014 kuta software llc. all rights reserved.
simplify.

1. $(8 + 7i)^2$
2. $(3 + 8i)(-2 - i)$
3. $(-1 + 6i)^2$
4. $(-1 - 8i)(6 - 5i)$
5. $(8 - 2i)^2$
6. $(8 + 5i)(-6 - 2i)$
7. $8(-2i)(-5 - 4i)$
8. $(2 - 2i)^2$
9. $(4i)(-2i)(-8 + 4i)$
10. $(-4 - 4i)(1 - 3i)$
11. $(7 - 2i)(6 - 4i)$
12. $(4 - 3i)(6 - 6i)$
13. $\frac{1 + 9i}{-2 + 9i}$
14. $\frac{-7 + 6i}{7 - 10i}$
15. $\frac{-5 - i}{-6 - 9i}$
16. $\frac{4 - 8i}{6 - 5i}$
17. $\frac{-1 + 3i}{-4 - 8i}$
18. $\frac{-4 + 4i}{4 - 6i}$
19. $\frac{-6 - i}{1 + 5i}$
20. $\frac{-4 + 3i}{-10 + 7i}$
21. $\frac{-2 + 3i}{3 - 2i}$
22. $\frac{-2 - 10i}{9 - 9i}$
23. $\frac{-7 + 6i}{10 + 5i}$
24. $\frac{-7 - 7i}{-7 - 4i}$

Explanation:

1) Step1: Expand using square formula

$(a+b)^2=a^2+2ab+b^2$
$\begin{align*}
(8+7i)^2&=8^2+2(8)(7i)+(7i)^2\\
&=64+112i+49i^2
\end{align*}$

1) Step2: Substitute $i^2=-1$

$\begin{align*}
&=64+112i+49(-1)\\
&=64-49+112i\\
&=15+112i
\end{align*}$

2) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$
$\begin{align*}
(3+8i)(-2-i)&=3(-2)+3(-i)+8i(-2)+8i(-i)\\
&=-6-3i-16i-8i^2
\end{align*}$

2) Step2: Combine like terms and substitute $i^2=-1$

$\begin{align*}
&=-6-19i-8(-1)\\
&=-6+8-19i\\
&=2-19i
\end{align*}$

3) Step1: Expand using square formula

$(a+b)^2=a^2+2ab+b^2$
$\begin{align*}
(-1+6i)^2&=(-1)^2+2(-1)(6i)+(6i)^2\\
&=1-12i+36i^2
\end{align*}$

3) Step2: Substitute $i^2=-1$

$\begin{align*}
&=1-12i+36(-1)\\
&=1-36-12i\\
&=-35-12i
\end{align*}$

4) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$
$\begin{align*}
(-1-8i)(6-5i)&=-1(6)-1(-5i)-8i(6)-8i(-5i)\\
&=-6+5i-48i+40i^2
\end{align*}$

4) Step2: Combine like terms and substitute $i^2=-1$

$\begin{align*}
&=-6-43i+40(-1)\\
&=-6-40-43i\\
&=-46-43i
\end{align*}$

5) Step1: Expand using square formula

$(a-b)^2=a^2-2ab+b^2$
$\begin{align*}
(8-2i)^2&=8^2-2(8)(2i)+(2i)^2\\
&=64-32i+4i^2
\end{align*}$

5) Step2: Substitute $i^2=-1$

$\begin{align*}
&=64-32i+4(-1)\\
&=64-4-32i\\
&=60-32i
\end{align*}$

6) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$
$\begin{align*}
(8+5i)(-6-2i)&=8(-6)+8(-2i)+5i(-6)+5i(-2i)\\
&=-48-16i-30i-10i^2
\end{align*}$

6) Step2: Combine like terms and substitute $i^2=-1$

$\begin{align*}
&=-48-46i-10(-1)\\
&=-48+10-46i\\
&=-38-46i
\end{align*}$

7) Step1: Multiply constants first

$8(-2i)=-16i$

7) Step2: Multiply with remaining complex number

$\begin{align*}
-16i(-5-4i)&=80i+64i^2
\end{align*}$

7) Step3: Substitute $i^2=-1$

$\begin{align*}
&=80i+64(-1)\\
&=-64+80i
\end{align*}$

8) Step1: Expand using square formula

$(a-b)^2=a^2-2ab+b^2$
$\begin{align*}
(2-2i)^2&=2^2-2(2)(2i)+(2i)^2\\
&=4-8i+4i^2
\end{align*}$

8) Step2: Substitute $i^2=-1$

$\begin{align*}
&=4-8i+4(-1)\\
&=4-4-8i\\
&=-8i
\end{align*}$

9) Step1: Multiply first two terms

$\begin{align*}
(4i)(-2i)&=-8i^2\\
&=-8(-1)=8
\end{align*}$

9) Step2: Multiply with remaining term

$\begin{align*}
8(-8+4i)&=-64+32i
\end{align*}$

10) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$
$\begin{align*}
(-4-4i)(1-3i)&=-4(1)-4(-3i)-4i(1)-4i(-3i)\\
&=-4+12i-4i+12i^2
\end{align*}$

10) Step2: Combine like terms and substitute $i^2=-1$

$\begin{align*}
&=-4+8i+12(-1)\\
&=-4-12+8i\\
&=-16+8i
\end{align*}$

11) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$
$\begin{align*}
(7-2i)(6-4i)&=7(6)+7(-4i)-2i(6)-2i(-4i)\\
&=42-28i-12i+8i^2
\end{align*}$

11) Step2: Combine like terms and substitute $i^2=-1$

$\begin{align*}
&=42-40i+8(-1)\\
&=42-8-40i\\
&=34-40i
\end{align*}$

12) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$
$\begin{align*}
(4-3i)(6-6i)&=4(6)+4(-6i)-3i(6)-3i(-6i)\\
&=24-24i-18i+18i^2
\end{align*}$

12) Step2: Combine like terms and substitute $i^2=-1$

$\begin{align*}
&=24-42i+18(-1)\\
&=24-18-42i\\
&=6-42i
\end{align*}$

13) Step1: Multiply by conjugate of denominator

Conjugate of $-2+9i$ is $-2-9i$
$\begin{align*}
\frac{1+9i}{-2+9i}&=\frac{(1+9i)(-2-9i)}{(-2+9i)(-2-9i)}
\end{align*}$

13) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=1(-2)+1(-9i)+9i(-2)+9i(-9i)\\
&=-2-9i-18i-81i^2=-2-27i-81(-1)=79-27i\\
\text{Denominator}&=(-2)^2-(9i)^2=4-81i^2=4-81(-1)=85
\end{align*}$

13) Step3: Write final form

$\frac{79-27i}{85}=\frac{79}{85}-\frac{27}{85}i$

14) Step1: Multiply by conjugate of denominator

Conjugate of $7-10i$ is $7+10i$
$\begin{align*}
\frac{-7+6i}{7-10i}&=\frac{(-7+6i)(7+10i)}{(7-10i)(7+10i)}
\end{align*}$

14) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=-7(7)-7(10i)+6i(7)+6i(10i)\\
&=-49-70i+42i+60i^2=-49-28i+60(-1)=-109-28i\\
\text{Denominator}&=7^2-(10i)^2=49-100i^2=49-100(-1)=149
\end{align*}$

14) Step3: Write final form

$\frac{-109-28i}{149}=-\frac{109}{149}-\frac{28}{149}i$

15) Step1: Multiply by conjugate of denominator

Conjugate of $-6-9i$ is $-6+9i$
$\begin{align*}
\frac{-5-i}{-6-9i}&=\frac{(-5-i)(-6+9i)}{(-6-9i)(-6+9i)}
\end{align*}$

15) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=-5(-6)-5(9i)-i(-6)-i(9i)\\
&=30-45i+6i-9i^2=30-39i-9(-1)=39-39i\\
\text{Denominator}&=(-6)^2-(9i)^2=36-81i^2=36-81(-1)=117
\end{align*}$

15) Step3: Simplify fraction

$\begin{align*}
\frac{39-39i}{117}&=\frac{39(1-i)}{39(3)}\\
&=\frac{1-i}{3}=\frac{1}{3}-\frac{1}{3}i
\end{align*}$

16) Step1: Multiply by conjugate of denominator

Conjugate of $6-5i$ is $6+5i$
$\begin{align*}
\frac{4-8i}{6-5i}&=\frac{(4-8i)(6+5i)}{(6-5i)(6+5i)}
\end{align*}$

16) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=4(6)+4(5i)-8i(6)-8i(5i)\\
&=24+20i-48i-40i^2=24-28i-40(-1)=64-28i\\
\text{Denominator}&=6^2-(5i)^2=36-25i^2=36-25(-1)=61
\end{align*}$

16) Step3: Write final form

$\frac{64-28i}{61}=\frac{64}{61}-\frac{28}{61}i$

17) Step1: Multiply by conjugate of denominator

Conjugate of $-4-8i$ is $-4+8i$
$\begin{align*}
\frac{-1+3i}{-4-8i}&=\frac{(-1+3i)(-4+8i)}{(-4-8i)(-4+8i)}
\end{align*}$

17) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=-1(-4)-1(8i)+3i(-4)+3i(8i)\\
&=4-8i-12i+24i^2=4-20i+24(-1)=-20-20i\\
\text{Denominator}&=(-4)^2-(8i)^2=16-64i^2=16-64(-1)=80
\end{align*}$

17) Step3: Simplify fraction

$\begin{align*}
\frac{-20-20i}{80}&=\frac{-20(1+i)}{20(4)}\\
&=-\frac{1+i}{4}=-\frac{1}{4}-\frac{1}{4}i
\end{align*}$

18) Step1: Multiply by conjugate of denominator

Conjugate of $4-6i$ is $4+6i$
$\begin{align*}
\frac{-4+4i}{4-6i}&=\frac{(-4+4i)(4+6i)}{(4-6i)(4+6i)}
\end{align*}$

18) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=-4(4)-4(6i)+4i(4)+4i(6i)\\
&=-16-24i+16i+24i^2=-16-8i+24(-1)=-40-8i\\
\text{Denominator}&=4^2-(6i)^2=16-36i^2=16-36(-1)=52
\end{align*}$

18) Step3: Simplify fraction

$\begin{align*}
\frac{-40-8i}{52}&=\frac{-4(10+2i)}{4(13)}\\
&=-\frac{10+2i}{13}=-\frac{10}{13}-\frac{2}{13}i
\end{align*}$

19) Step1: Multiply by conjugate of denominator

Conjugate of $1+5i$ is $1-5i$
$\begin{align*}
\frac{-6-i}{1+5i}&=\frac{(-6-i)(1-5i)}{(1+5i)(1-5i)}
\end{align*}$

19) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=-6(1)-6(-5i)-i(1)-i(-5i)\\
&=-6+30i-i+5i^2=-6+29i+5(-1)=-11+29i\\
\text{Denominator}&=1^2-(5i)^2=1-25i^2=1-25(-1)=26
\end{align*}$

19) Step3: Write final form

$\frac{-11+29i}{26}=-\frac{11}{26}+\frac{29}{26}i$

20) Step1: Multiply by conjugate of denominator

Conjugate of $-10+7i$ is $-10-7i$
$\begin{align*}
\frac{-4+3i}{-10+7i}&=\frac{(-4+3i)(-10-7i)}{(-10+7i)(-10-7i)}
\end{align*}$

20) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=-4(-10)-4(-7i)+3i(-10)+3i(-7i)\\
&=40+28i-30i-21i^2=40-2i-21(-1)=61-2i\\
\text{Denominator}&=(-10)^2-(7i)^2=100-49i^2=100-49(-1)=149
\end{align*}$

20) Step3: Write final form

$\frac{61-2i}{149}=\frac{61}{149}-\frac{2}{149}i$

21) Step1: Multiply by conjugate of denominator

Conjugate of $3-2i$ is $3+2i$
$\begin{align*}
\frac{-2+3i}{3-2i}&=\frac{(-2+3i)(3+2i)}{(3-2i)(3+2i)}
\end{align*}$

21) Step2: Expand numerator and denominator

$\begin{align*}
\text{Numerator}&=-2(3)-2(2i)+3i(3)+3i(2i)\\
&=-6-4i+9i+6i^2=-6+5i+6(-1)=-12+5i\\
\text{Denominator}&=3^2-(2i)^2=9-4i^2=9-4(-1)=13
\end{align*}$

Answer:

1) Step1: Expand using square formula

$(a+b)^2=a^2+2ab+b^2$

$$\begin{align*} (8+7i)^2&=8^2+2(8)(7i)+(7i)^2\\ &=64+112i+49i^2 \end{align*}$$

1) Step2: Substitute $i^2=-1$

$$\begin{align*} &=64+112i+49(-1)\\ &=64-49+112i\\ &=15+112i \end{align*}$$

2) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$

$$\begin{align*} (3+8i)(-2-i)&=3(-2)+3(-i)+8i(-2)+8i(-i)\\ &=-6-3i-16i-8i^2 \end{align*}$$

2) Step2: Combine like terms and substitute $i^2=-1$

$$\begin{align*} &=-6-19i-8(-1)\\ &=-6+8-19i\\ &=2-19i \end{align*}$$

3) Step1: Expand using square formula

$(a+b)^2=a^2+2ab+b^2$

$$\begin{align*} (-1+6i)^2&=(-1)^2+2(-1)(6i)+(6i)^2\\ &=1-12i+36i^2 \end{align*}$$

3) Step2: Substitute $i^2=-1$

$$\begin{align*} &=1-12i+36(-1)\\ &=1-36-12i\\ &=-35-12i \end{align*}$$

4) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$

$$\begin{align*} (-1-8i)(6-5i)&=-1(6)-1(-5i)-8i(6)-8i(-5i)\\ &=-6+5i-48i+40i^2 \end{align*}$$

4) Step2: Combine like terms and substitute $i^2=-1$

$$\begin{align*} &=-6-43i+40(-1)\\ &=-6-40-43i\\ &=-46-43i \end{align*}$$

5) Step1: Expand using square formula

$(a-b)^2=a^2-2ab+b^2$

$$\begin{align*} (8-2i)^2&=8^2-2(8)(2i)+(2i)^2\\ &=64-32i+4i^2 \end{align*}$$

5) Step2: Substitute $i^2=-1$

$$\begin{align*} &=64-32i+4(-1)\\ &=64-4-32i\\ &=60-32i \end{align*}$$

6) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$

$$\begin{align*} (8+5i)(-6-2i)&=8(-6)+8(-2i)+5i(-6)+5i(-2i)\\ &=-48-16i-30i-10i^2 \end{align*}$$

6) Step2: Combine like terms and substitute $i^2=-1$

$$\begin{align*} &=-48-46i-10(-1)\\ &=-48+10-46i\\ &=-38-46i \end{align*}$$

7) Step1: Multiply constants first

$8(-2i)=-16i$

7) Step2: Multiply with remaining complex number

$$\begin{align*} -16i(-5-4i)&=80i+64i^2 \end{align*}$$

7) Step3: Substitute $i^2=-1$

$$\begin{align*} &=80i+64(-1)\\ &=-64+80i \end{align*}$$

8) Step1: Expand using square formula

$(a-b)^2=a^2-2ab+b^2$

$$\begin{align*} (2-2i)^2&=2^2-2(2)(2i)+(2i)^2\\ &=4-8i+4i^2 \end{align*}$$

8) Step2: Substitute $i^2=-1$

$$\begin{align*} &=4-8i+4(-1)\\ &=4-4-8i\\ &=-8i \end{align*}$$

9) Step1: Multiply first two terms

$$\begin{align*} (4i)(-2i)&=-8i^2\\ &=-8(-1)=8 \end{align*}$$

9) Step2: Multiply with remaining term

$$\begin{align*} 8(-8+4i)&=-64+32i \end{align*}$$

10) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$

$$\begin{align*} (-4-4i)(1-3i)&=-4(1)-4(-3i)-4i(1)-4i(-3i)\\ &=-4+12i-4i+12i^2 \end{align*}$$

10) Step2: Combine like terms and substitute $i^2=-1$

$$\begin{align*} &=-4+8i+12(-1)\\ &=-4-12+8i\\ &=-16+8i \end{align*}$$

11) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$

$$\begin{align*} (7-2i)(6-4i)&=7(6)+7(-4i)-2i(6)-2i(-4i)\\ &=42-28i-12i+8i^2 \end{align*}$$

11) Step2: Combine like terms and substitute $i^2=-1$

$$\begin{align*} &=42-40i+8(-1)\\ &=42-8-40i\\ &=34-40i \end{align*}$$

12) Step1: Expand using FOIL method

$(a+b)(c+d)=ac+ad+bc+bd$

$$\begin{align*} (4-3i)(6-6i)&=4(6)+4(-6i)-3i(6)-3i(-6i)\\ &=24-24i-18i+18i^2 \end{align*}$$

12) Step2: Combine like terms and substitute $i^2=-1$

$$\begin{align*} &=24-42i+18(-1)\\ &=24-18-42i\\ &=6-42i \end{align*}$$

13) Step1: Multiply by conjugate of denominator

Conjugate of $-2+9i$ is $-2-9i$

$$\begin{align*} \frac{1+9i}{-2+9i}&=\frac{(1+9i)(-2-9i)}{(-2+9i)(-2-9i)} \end{align*}$$

13) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=1(-2)+1(-9i)+9i(-2)+9i(-9i)\\ &=-2-9i-18i-81i^2=-2-27i-81(-1)=79-27i\\ \text{Denominator}&=(-2)^2-(9i)^2=4-81i^2=4-81(-1)=85 \end{align*}$$

13) Step3: Write final form

$\frac{79-27i}{85}=\frac{79}{85}-\frac{27}{85}i$

14) Step1: Multiply by conjugate of denominator

Conjugate of $7-10i$ is $7+10i$

$$\begin{align*} \frac{-7+6i}{7-10i}&=\frac{(-7+6i)(7+10i)}{(7-10i)(7+10i)} \end{align*}$$

14) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=-7(7)-7(10i)+6i(7)+6i(10i)\\ &=-49-70i+42i+60i^2=-49-28i+60(-1)=-109-28i\\ \text{Denominator}&=7^2-(10i)^2=49-100i^2=49-100(-1)=149 \end{align*}$$

14) Step3: Write final form

$\frac{-109-28i}{149}=-\frac{109}{149}-\frac{28}{149}i$

15) Step1: Multiply by conjugate of denominator

Conjugate of $-6-9i$ is $-6+9i$

$$\begin{align*} \frac{-5-i}{-6-9i}&=\frac{(-5-i)(-6+9i)}{(-6-9i)(-6+9i)} \end{align*}$$

15) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=-5(-6)-5(9i)-i(-6)-i(9i)\\ &=30-45i+6i-9i^2=30-39i-9(-1)=39-39i\\ \text{Denominator}&=(-6)^2-(9i)^2=36-81i^2=36-81(-1)=117 \end{align*}$$

15) Step3: Simplify fraction

$$\begin{align*} \frac{39-39i}{117}&=\frac{39(1-i)}{39(3)}\\ &=\frac{1-i}{3}=\frac{1}{3}-\frac{1}{3}i \end{align*}$$

16) Step1: Multiply by conjugate of denominator

Conjugate of $6-5i$ is $6+5i$

$$\begin{align*} \frac{4-8i}{6-5i}&=\frac{(4-8i)(6+5i)}{(6-5i)(6+5i)} \end{align*}$$

16) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=4(6)+4(5i)-8i(6)-8i(5i)\\ &=24+20i-48i-40i^2=24-28i-40(-1)=64-28i\\ \text{Denominator}&=6^2-(5i)^2=36-25i^2=36-25(-1)=61 \end{align*}$$

16) Step3: Write final form

$\frac{64-28i}{61}=\frac{64}{61}-\frac{28}{61}i$

17) Step1: Multiply by conjugate of denominator

Conjugate of $-4-8i$ is $-4+8i$

$$\begin{align*} \frac{-1+3i}{-4-8i}&=\frac{(-1+3i)(-4+8i)}{(-4-8i)(-4+8i)} \end{align*}$$

17) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=-1(-4)-1(8i)+3i(-4)+3i(8i)\\ &=4-8i-12i+24i^2=4-20i+24(-1)=-20-20i\\ \text{Denominator}&=(-4)^2-(8i)^2=16-64i^2=16-64(-1)=80 \end{align*}$$

17) Step3: Simplify fraction

$$\begin{align*} \frac{-20-20i}{80}&=\frac{-20(1+i)}{20(4)}\\ &=-\frac{1+i}{4}=-\frac{1}{4}-\frac{1}{4}i \end{align*}$$

18) Step1: Multiply by conjugate of denominator

Conjugate of $4-6i$ is $4+6i$

$$\begin{align*} \frac{-4+4i}{4-6i}&=\frac{(-4+4i)(4+6i)}{(4-6i)(4+6i)} \end{align*}$$

18) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=-4(4)-4(6i)+4i(4)+4i(6i)\\ &=-16-24i+16i+24i^2=-16-8i+24(-1)=-40-8i\\ \text{Denominator}&=4^2-(6i)^2=16-36i^2=16-36(-1)=52 \end{align*}$$

18) Step3: Simplify fraction

$$\begin{align*} \frac{-40-8i}{52}&=\frac{-4(10+2i)}{4(13)}\\ &=-\frac{10+2i}{13}=-\frac{10}{13}-\frac{2}{13}i \end{align*}$$

19) Step1: Multiply by conjugate of denominator

Conjugate of $1+5i$ is $1-5i$

$$\begin{align*} \frac{-6-i}{1+5i}&=\frac{(-6-i)(1-5i)}{(1+5i)(1-5i)} \end{align*}$$

19) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=-6(1)-6(-5i)-i(1)-i(-5i)\\ &=-6+30i-i+5i^2=-6+29i+5(-1)=-11+29i\\ \text{Denominator}&=1^2-(5i)^2=1-25i^2=1-25(-1)=26 \end{align*}$$

19) Step3: Write final form

$\frac{-11+29i}{26}=-\frac{11}{26}+\frac{29}{26}i$

20) Step1: Multiply by conjugate of denominator

Conjugate of $-10+7i$ is $-10-7i$

$$\begin{align*} \frac{-4+3i}{-10+7i}&=\frac{(-4+3i)(-10-7i)}{(-10+7i)(-10-7i)} \end{align*}$$

20) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=-4(-10)-4(-7i)+3i(-10)+3i(-7i)\\ &=40+28i-30i-21i^2=40-2i-21(-1)=61-2i\\ \text{Denominator}&=(-10)^2-(7i)^2=100-49i^2=100-49(-1)=149 \end{align*}$$

20) Step3: Write final form

$\frac{61-2i}{149}=\frac{61}{149}-\frac{2}{149}i$

21) Step1: Multiply by conjugate of denominator

Conjugate of $3-2i$ is $3+2i$

$$\begin{align*} \frac{-2+3i}{3-2i}&=\frac{(-2+3i)(3+2i)}{(3-2i)(3+2i)} \end{align*}$$

21) Step2: Expand numerator and denominator

$$\begin{align*} \text{Numerator}&=-2(3)-2(2i)+3i(3)+3i(2i)\\ &=-6-4i+9i+6i^2=-6+5i+6(-1)=-12+5i\\ \text{Denominator}&=3^2-(2i)^2=9-4i^2=9-4(-1)=13 \end{align*}$$