SOLUTION: Worksheet on operations on complex numbers - Studypool - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

(1 – 6) Add and subtract

*Rule: Combine the real numbers together and the imaginary numbers ($i$) together.*

1) $(4 + i) + (7 - i)$
* Real parts: $4 + 7 = 11$
* Imaginary parts: $1i - 1i = 0$
* Answer: $11$

2) $(-4 + 3i) - (1 - 8i)$
* Distribute the negative sign: $-4 + 3i - 1 + 8i$
* Real parts: $-4 - 1 = -5$
* Imaginary parts: $3i + 8i = 11i$
* Answer: $-5 + 11i$

3) $(-5 + 3i) - (-2 + 6i)$
* Distribute the negative sign: $-5 + 3i + 2 - 6i$
* Real parts: $-5 + 2 = -3$
* Imaginary parts: $3i - 6i = -3i$
* Answer: $-3 - 3i$

4) $(-3 + 8i) + (5 + 6i)$
* Real parts: $-3 + 5 = 2$
* Imaginary parts: $8i + 6i = 14i$
* Answer: $2 + 14i$

5) $(-2i) - 7 - (-2 + 8i)$
* Rewrite in order: $-7 - 2i - (-2 + 8i)$
* Distribute the negative sign: $-7 - 2i + 2 - 8i$
* Real parts: $-7 + 2 = -5$
* Imaginary parts: $-2i - 8i = -10i$
* Answer: $-5 - 10i$

6) $(5 - 7i) - (2 + i)$
* Distribute the negative sign: $5 - 7i - 2 - 1i$
* Real parts: $5 - 2 = 3$
* Imaginary parts: $-7i - 1i = -8i$
* Answer: $3 - 8i$

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(7 – 9) Simplify and write in standard form

*Rule: Use FOIL (First, Outer, Inner, Last) to multiply. Remember that $i^2 = -1$.*

7) $5(-8i)(1 + i)$
* First, multiply the single terms: $5 \times -8i = -40i$
* Now multiply by the parenthesis: $-40i(1 + i)$
* Distribute: $-40i(1) + (-40i)(i) = -40i - 40i^2$
* Substitute $i^2 = -1$: $-40i - 40(-1)$
* Simplify: $-40i + 40$
* Standard form ($a+bi$): $40 - 40i$

8) $(-8 - 8i)^2$
* This means $(-8 - 8i)(-8 - 8i)$
* First: $(-8)(-8) = 64$
* Outer: $(-8)(-8i) = +64i$
* Inner: $(-8i)(-8) = +64i$
* Last: $(-8i)(-8i) = +64i^2$
* Combine: $64 + 128i + 64i^2$
* Substitute $i^2 = -1$: $64 + 128i + 64(-1)$
* Simplify: $64 + 128i - 64$
* The real parts cancel out ($64-64=0$).
* Answer: $128i$

9) $(-4 + 2i)^2$
* This means $(-4 + 2i)(-4 + 2i)$
* First: $(-4)(-4) = 16$
* Outer: $(-4)(2i) = -8i$
* Inner: $(2i)(-4) = -8i$
* Last: $(2i)(2i) = 4i^2$
* Combine: $16 - 16i + 4i^2$
* Substitute $i^2 = -1$: $16 - 16i + 4(-1)$
* Simplify: $16 - 16i - 4$
* Combine reals: $12 - 16i$
* Answer: $12 - 16i$

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(10 - 13) Challenge problems

*Rule: Follow order of operations. Do powers first, then multiplication, then addition/subtraction.*

10) $-6 - (-2 - 2i) - (5 - 4i)$
* Distribute negatives: $-6 + 2 + 2i - 5 + 4i$
* Real parts: $-6 + 2 - 5 = -9$
* Imaginary parts: $2i + 4i = 6i$
* Answer: $-9 + 6i$

11) $-4(-7 + 8i)(-5 + 6i)$
* First, multiply the two binomials $(-7 + 8i)(-5 + 6i)$:
* $(-7)(-5) = 35$
* $(-7)(6i) = -42i$
* $(8i)(-5) = -40i$
* $(8i)(6i) = 48i^2 = 48(-1) = -48$
* Combine: $35 - 48 - 42i - 40i = -13 - 82i$
* Now multiply by the $-4$ in front:
* $-4(-13 - 82i)$
* $(-4)(-13) = 52$
* $(-4)(-82i) = +328i$
* Answer: $52 + 328i$

12) $(-5 + 5i) - (4 - 2i) + (-8 - 7i)^2$
* Step 1: Expand the square $(-8 - 7i)^2$
* $(-8)(-8) = 64$
* $2(-8)(-7i) = 112i$
* $(-7i)(-7i) = 49i^2 = -49$
* Result: $64 - 49 + 112i = 15 + 112i$
* Step 2: Put it all back into the equation
* $(-5 + 5i) - (4 - 2i) + (15 + 112i)$
* Distribute the middle negative: $-5 + 5i - 4 + 2i + 15 + 112i$
* Step 3: Combine like terms
* Real: $-5 - 4 + 15 = 6$
* Imaginary: $5i + 2i + 112i = 119i$
* Answer: $6 + 119i$

13) $(-4 - 6i) - (-7 - i) + (-5 - 4i)(8 + 4i)$
* Step 1: Multiply $(-5 - 4i)(8 + 4i)$
* $(-5)(8) = -40$
* $(-5)(4i) = -20i$
* $(-4i)(8) = -32i$
* $(-4i)(4i) = -16i^2 = -16(-1) = +16$
* Combine: $-40 + 16 - 20i - 32i = -24 - 52i$
* Step 2: Put it all back into the equation
* $(-4 - 6i) - (-7 - i) + (-24 - 52i)$
* Distribute the middle negative: $-4 - 6i + 7 + i - 24 - 52i$
* Step 3: Combine like terms
* Real: $-4 + 7 - 24 = -21$
* Imaginary: $-6i + 1i - 52i = -57i$
* Answer: $-21 - 57i$

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(14 – 16) Perform division

*Rule: Multiply the top and bottom by the "conjugate" of the denominator. The conjugate changes the sign in the middle (e.g., conjugate of $-3+i$ is $-3-i$).*

14) $\frac{3i}{-3 + i}$
* Conjugate of bottom: $-3 - i$
* Top: $3i(-3 - i) = -9i - 3i^2 = -9i - 3(-1) = 3 - 9i$
* Bottom: $(-3 + i)(-3 - i) = 9 - i^2 = 9 - (-1) = 10$
* Result: $\frac{3 - 9i}{10}$
* Answer: $\frac{3}{10} - \frac{9}{10}i$

15) $\frac{10i}{-6 + i}$
* Conjugate of bottom: $-6 - i$
* Top: $10i(-6 - i) = -60i - 10i^2 = -60i - 10(-1) = 10 - 60i$
* Bottom: $(-6 + i)(-6 - i) = 36 - i^2 = 36 - (-1) = 37$
* Result: $\frac{10 - 60i}{37}$
* Answer: $\frac{10}{37} - \frac{60}{37}i$

16) $\frac{5}{6 - 2i}$
* Conjugate of bottom: $6 + 2i$
* Top: $5(6 + 2i) = 30 + 10i$
* Bottom: $(6 - 2i)(6 + 2i) = 36 - 4i^2 = 36 - 4(-1) = 40$
* Result: $\frac{30 + 10i}{40}$
* Simplify fractions (divide by 10): $\frac{3 + i}{4}$
* Answer: $\frac{3}{4} + \frac{1}{4}i$

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(17 – 20) Simplify

*Same rule as above: Multiply by the conjugate.*

17) $\frac{-7 - 3i}{2 + 6i}$
* Conjugate of bottom: $2 - 6i$
* Top: $(-7 - 3i)(2 - 6i)$
* $-14 + 42i - 6i + 18i^2$
* $-14 + 36i + 18(-1)$
* $-14 - 18 + 36i = -32 + 36i$
* Bottom: $(2 + 6i)(2 - 6i) = 4 - 36i^2 = 4 - 36(-1) = 40$
* Result: $\frac{-32 + 36i}{40}$
* Simplify (divide by 4): $\frac{-8 + 9i}{10}$
* Answer: $-\frac{8}{10} + \frac{9}{10}i$ (or $-\frac{4}{5} + \frac{9}{10}i$)

18) $\frac{4 + i}{2 + 6i}$
* Conjugate of bottom: $2 - 6i$
* Top: $(4 + i)(2 - 6i)$
* $8 - 24i + 2i - 6i^2$
* $8 - 22i - 6(-1)$
* $8 + 6 - 22i = 14 - 22i$
* Bottom: $(2 + 6i)(2 - 6i) = 4 - 36i^2 = 40$
* Result: $\frac{14 - 22i}{40}$
* Simplify (divide by 2): $\frac{7 - 11i}{20}$
* Answer: $\frac{7}{20} - \frac{11}{20}i$

19) $\frac{7 - 7i}{7 - 5i}$
* Conjugate of bottom: $7 + 5i$
* Top: $(7 - 7i)(7 + 5i)$
* $49 + 35i - 49i - 35i^2$
* $49 - 14i - 35(-1)$
* $49 + 35 - 14i = 84 - 14i$
* Bottom: $(7 - 5i)(7 + 5i) = 49 - 25i^2 = 49 - 25(-1) = 74$
* Result: $\frac{84 - 14i}{74}$
* Simplify (divide by 2): $\frac{42 - 7i}{37}$
* Answer: $\frac{42}{37} - \frac{7}{37}i$

20) $\frac{-10 - 2i}{-8 + 6i}$
* Conjugate of bottom: $-8 - 6i$
* Top: $(-10 - 2i)(-8 - 6i)$
* $80 + 60i + 16i + 12i^2$
* $80 + 76i + 12(-1)$
* $80 - 12 + 76i = 68 + 76i$
* Bottom: $(-8 + 6i)(-8 - 6i) = 64 - 36i^2 = 64 - 36(-1) = 100$
* Result: $\frac{68 + 76i}{100}$
* Simplify (divide by 4): $\frac{17 + 19i}{25}$
* Answer: $\frac{17}{25} + \frac{19}{25}i$

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Final Answer:
1) $11$
2) $-5 + 11i$
3) $-3 - 3i$
4) $2 + 14i$
5) $-5 - 10i$
6) $3 - 8i$
7) $40 - 40i$
8) $128i$
9) $12 - 16i$
10) $-9 + 6i$
11) $52 + 328i$
12) $6 + 119i$
13) $-21 - 57i$
14) $\frac{3}{10} - \frac{9}{10}i$
15) $\frac{10}{37} - \frac{60}{37}i$
16) $\frac{3}{4} + \frac{1}{4}i$
17) $-\frac{4}{5} + \frac{9}{10}i$
18) $\frac{7}{20} - \frac{11}{20}i$
19) $\frac{42}{37} - \frac{7}{37}i$
20) $\frac{17}{25} + \frac{19}{25}i$