Question

Perform the indicated operation and write your answer in standard form.

1. (4-5i)(12+11i)(4-5i)(12+11i)

2. (-3-1)-(6-7i)(-3-i)-(6-7i) 3. (1+4i)-(-16+91)(1+41)-(-16+91)

4. 8i(10+2i)8i(10+21)

5. (-3-9i)(1+10i)(-3-91)(1+10i)

6. (2+7i)(8+3i)(2+7i)(8+3i)

7. (3-21)-(8-51)

8. (4-2)*(1-Si)

9. (-2-41)/1 10: (-3+2)/(3-6)

11. If ix-yi)/i (7+9i).where x and y are real, what is the value of (x+ yikx-yi)?

12. Determine all complex number z that satisfy the equation

2+32-5-61.

13. If zel+i, find 2²

14. Find all complex numbers of the form zabi, where a and b are real numbers such that 22 25 and a+b=7. 15. Find all complex nunibers z such that (4+2ijz+ (8-2) z--2-101.

16 write the following standard form

Answer 1

Answer:

(4-5i)(12+11i)(4-5i)(12+11i)

Answer 2

Answer:

Step-by-step explanation:

1.(4−5i)(4−5i)

Multiply complex numbers 4−5i and 4−5i like you multiply binomials.

4×4+4×(−5i)−5i×4−5(−5)i2

By definition, i2  is −1.

4×4+4×(−5i)−5i×4−5(−5)(−1)

Do the multiplications.

16−20i−20i−25

Combine the real and imaginary parts.

16−25+(−20−20)i

Do the additions.

−9−40i

REAL PART

−9

(12+11i)(12+11i)

Multiply complex numbers 12+11i and 12+11i like you multiply binomials.

12×12+12×(11i)+11i×12+11×11i2

By definition, i2  is −1.

12×12+12×(11i)+11i×12+11×11(−1)

144+132i+132i−121

144−121+(132+132)i

23+264i

REAL PART

23

2.−3−i−(6−7i)(−3−i)−6−7i

Multiply complex numbers 6−7i and −3−i like you multiply binomials.

−3−i−(6(−3)+6(−i)−7i(−3)−7(−1)i

2

)−6−7i

By definition, i

2

 is −1.

−3−i−(6(−3)+6(−i)−7i(−3)−7(−1)(−1))−6−7i

Do the multiplications in 6(−3)+6(−i)−7i(−3)−7(−1)(−1).

−3−i−(−18−6i+21i−7)−6−7i

Combine the real and imaginary parts in −18−6i+21i−7.

−3−i−(−18−7+(−6+21)i)−6−7i

Do the additions in −18−7+(−6+21)i.

−3−i−(−25+15i)−6−7i

Subtract −25+15i from −3−i by subtracting corresponding real and imaginary parts.

−3−(−25)+(−1−15)i−6−7i

Subtract −25 from −3. Subtract 15 from −1.

22−16i−6−7i

22−6+(−16−7)i

16−23i

REAL PART

16

3.1+4i−(−16+9i)(1+41)−(−16+9i)

Add 1 and 41 to get 42.

1+4i−(−16+9i)×42−(−16+9i)

Multiply −16+9i times 42.

1+4i−(−16×42+9i×42)−(−16+9i)

Do the multiplications in −16×42+9i×42.

1+4i−(−672+378i)−(−16+9i)

Subtract −672+378i from 1+4i by subtracting corresponding real and imaginary parts.

1−(−672)+(4−378)i−(−16+9i)

Subtract −672 from 1. Subtract 378 from 4.

673−374i−(−16+9i)

Subtract −16+9i from 673−374i by subtracting corresponding real and imaginary parts.

673−(−16)+(−374−9)i

Subtract −16 from 673. Subtract 9 from −374.

689−383i

REAL PART

689

4.8i(10+2i)8i(10+2i)

Multiply 8i times 10+2i.

(8i×10+8×2i2)×(8i)(10+2i)

By definition, i2  is −1.

(8i×10+8×2(−1))×(8i)(10+2i)

(−16+80i)×(8i)(10+2i)

Multiply −16+80i times 8i.

(−16×(8i)+80×8i2 )(10+2i)

By definition, i2  is −1.

(−16×(8i)+80×8(−1))(10+2i)

(−640−128i)(10+2i)

Multiply complex numbers −640−128i and 10+2i like you multiply binomials.

−640×10−640×(2i)−128i×10−128×2i2

By definition, i2  is −1.

−640×10−640×(2i)−128i×10−128×2(−1)

−6400−1280i−1280i+256

−6400+256+(−1280−1280)i

−6144−2560i

REAL PART

−6144

5.

(1+10i)(1+10i)

Multiply complex numbers 1+10i and 1+10i like you multiply binomials.

1×1+1×(10i)+10i×1+10×10i2

By definition, i2  is −1.

1×1+1×(10i)+10i×1+10×10(−1)

1+10i+10i−100

1−100+(10+10)i

−99+20i

REAL PART

−99

(−3−9i)(−3−9i)

Multiply complex numbers −3−9i and −3−9i like you multiply binomials.

−3(−3)−3×(−9i)−9i(−3)−9(−9)i

2

By definition, i 2

 is −1.

−3(−3)−3×(−9i)−9i(−3)−9(−9)(−1)

9+27i+27i−81

9−81+(27+27)i

−72+54i

REAL PART

−72

=-171

6.Expand (2+7i)(2+7i) using the FOIL Method.

Apply the distributive property.

$2(2+7i)+7i(2+7i)$

Apply the distributive property.

$2*2+2(7i)+7i(2*7i)$

Apply the distributive property.

$2*2+2(7i)+7i*2+7i(7i)$

Simplify and combine like terms.

$2*2+2(7i)+7i*2+7i(7i)$

$2*2+14i+7i*2+7i(7i)$

$4+14i+14i+7i(7i)$

$4+28i+49(i^2)\\4+28i-49\\28i-45$$(i^2)$

Expand (8+3i)(8+3i) using the FOIL Method.

Apply the distributive property.

$8(8+3i)+3i(8+3i)$

Apply the distributive property.

$8(8+3i)+3i(8+3i)\\64+24i+24i+9i^2 == > i^2=-1\\64+48i+9(-1)\\64-9+48i\\55+48i$

$(28i-45)(55+48i)$

Use the distributive property to multiply 28i−45 by 55+48i.

$-1344+1540i+(-2475−2160i)$

Combine the real and imaginary parts in numbers −1344+1540i and −2475−2160i.

−1344−2475+(1540−2160)i

Add −1344 to −2475. Add 1540 to −2160.

−3819−620i

REAL PART

−3819

7) -5 + 7i

8) -6 – 22i

9) -4 + 2i

10) -7/15 – 4i/15

11. (x + yi) / i = ( 7 + 9i )

(x + yi) = i(7 + 9i) = -9 + 7i

(x + yi)(x - yi) = (-9 + 7i)(-9 - 7i) = 81 + 49 = 130

12. Let z = a + bi , z' = a - bi ; a and b real numbers.

Substituting z and z' in the given equation obtain

a + bi + 3*(a - bi) = 5 - 6i

a + 3a + (b - 3b) i = 5 - 6i

4a = 5 and -2b = -6

a = 5/4 and b = 3

z = 5/4 + 3i

14.z z' = (a + bi)(a - bi)

= a2 + b2 = 25

a + b = 7 gives b = 7 - a

Substitute above in the equation a2 + b2 = 25

a2 + (7 - a)2 = 25

Solve the above quadratic function for a and use b = 7 - a to find b.

a = 4 and b = 3 or a = 3 and b = 4

z = 4 + 3i and z = 3 + 4i have the property z z' = 25.

15.Let z = a + bi where a and b are real numbers. The complex conjugate z' is written in terms of a and b as follows: z'= a - bi. Substitute z and z' in the given equation

(4 + 2i)(a + bi) + (8 - 2i)(a - bi) = -2 + 10i

Expand and separate real and imaginary parts.

(4a - 2b + 8a - 2b) + (4b + 2a - 8b - 2a )i = -2 + 10i

Two complex numbers are equal if their real parts and imaginary parts are equal. Group like terms.

12a - 4b = -2 and - 4b = 10

Solve the system of the unknown a and b to find:

b = -5/2 and a = -1

z = -1 - (5/2)i

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