Question

unit 20 day 01 homework: complex numbers practice
1 - 18: simplify each operation completely.

1. (-4 + 2i)+(6 - 3i)
2. (5 - i)-(3 - 2i)
3. (2 + i)(3 - i)
4. (5 - 2i)(4 - i)
5. (4 + 2i)(4 - 2i)
6. (-3i)(4i)(-5i)
7. (7i)^2(6i)
8. (4 + 3i)(2 - 5i)
9. \frac{2i}{4 + 2i}
10. \frac{2}{7 - 8i}
11. \frac{3 - i}{2 - i}
12. \frac{8 - 6i}{3i}
13. i^{55}
14. i^{89}
15. i^{42}
16. \sqrt{-36}
17. \sqrt{-8}\cdot\sqrt{-32}
18. \sqrt{-108x^{7}}

19 - 24: solve each equation.

1. 3x^{2}+3 = 0
2. 5x^{2}+35 = 0
3. 4x^{2}+120 = 0
4. x^{2}-16 = 0
5. 8x^{2}+96 = 0
6. \frac{3}{4}x^{2}+12 = 0

fact: complex numbers are used with electricity. in these problems, j usually represents the imaginary unit to avoid confusion with the i as a variable for current. using the formula e = iz, find the voltage v in a circuit when the current i is 3 - j amps and the impedance z is...

Explanation:

Step1: Simplify complex - number addition

For \((-4 + 2i)+(6 - 3i)\), combine real and imaginary parts: \((-4 + 6)+(2i-3i)=2 - i\)

Step2: Simplify complex - number subtraction

For \((5 - i)-(3 - 2i)\), distribute the negative sign and combine: \((5 - 3)+(-i + 2i)=2 + i\)

Step3: Expand complex - number multiplication

For \((2 + i)(3 - i)\), use FOIL method: \(2\times3-2i+3i - i^{2}=6 + i+1=7 + i\)

Step4: Expand complex - number multiplication

For \((5 - 2i)(4 - i)\), \(5\times4-5i-2i\times4+2i^{2}=20-5i - 8i-2 = 18-13i\)

Step5: Use the difference - of - squares formula for complex numbers

For \((4 + 2i)(4 - 2i)\), since \((a + bi)(a - bi)=a^{2}+b^{2}\), here \(a = 4\) and \(b = 2\), so \(4^{2}+2^{2}=16 + 4=20\)

Step6: Multiply complex numbers

For \((-3i)(4i)(-5i)\), first \((-3i)(4i)=-12i^{2}=12\), then \(12\times(-5i)=-60i\)

Step7: Calculate powers of \(i\) and multiply

For \((7i)^{2}(6i)\), \((7i)^{2}=49i^{2}=-49\), then \(-49\times6i=-294i\)

Step8: Expand complex - number multiplication

For \((4 + 3i)(2 - 5i)\), \(4\times2-4\times5i+3i\times2-3i\times5i=8-20i + 6i+15=23-14i\)

Step9: Rationalize the denominator

For \(\frac{2i}{4 + 2i}\), multiply numerator and denominator by the conjugate \(4 - 2i\): \(\frac{2i(4 - 2i)}{(4 + 2i)(4 - 2i)}=\frac{8i-4i^{2}}{16+4}=\frac{4 + 8i}{20}=\frac{1}{5}+\frac{2}{5}i\)

Step10: Rationalize the denominator

For \(\frac{2}{7 - 8i}\), multiply numerator and denominator by the conjugate \(7 + 8i\): \(\frac{2(7 + 8i)}{(7 - 8i)(7 + 8i)}=\frac{14 + 16i}{49 + 64}=\frac{14}{113}+\frac{16}{113}i\)

Step11: Rationalize the denominator

For \(\frac{3 - i}{2 - i}\), multiply numerator and denominator by the conjugate \(2 + i\): \(\frac{(3 - i)(2 + i)}{(2 - i)(2 + i)}=\frac{6+3i-2i - i^{2}}{4 + 1}=\frac{7 + i}{5}=\frac{7}{5}+\frac{1}{5}i\)

Step12: Simplify the complex - number fraction

For \(\frac{8 - 6i}{3i}\), multiply numerator and denominator by \(i\): \(\frac{(8 - 6i)i}{3i^{2}}=\frac{8i-6i^{2}}{-3}=\frac{6 + 8i}{-3}=-2-\frac{8}{3}i\)

Step13: Calculate powers of \(i\)

For \(i^{55}\), since \(i^{1}=i\), \(i^{2}=-1\), \(i^{3}=-i\), \(i^{4}=1\), and \(55 = 4\times13+3\), so \(i^{55}=i^{3}=-i\)

Step14: Calculate powers of \(i\)

For \(i^{89}\), since \(89 = 4\times22+1\), so \(i^{89}=i\)

Step15: Calculate powers of \(i\)

For \(i^{42}\), since \(42 = 4\times10+2\), so \(i^{42}=i^{2}=-1\)

Step16: Simplify square - root of a negative number

For \(\sqrt{-36}\), \(\sqrt{-36}=\sqrt{36}\times\sqrt{-1}=6i\)

Step17: Simplify square - root of negative numbers

For \(\sqrt{-8}\cdot\sqrt{-32}\), \(\sqrt{-8}=2\sqrt{2}i\), \(\sqrt{-32}=4\sqrt{2}i\), then \(2\sqrt{2}i\times4\sqrt{2}i = 16i^{2}=-16\)

Step18: Simplify square - root of a negative number with a variable

For \(\sqrt{-108x^{7}}\), \(\sqrt{-108x^{7}}=\sqrt{108}\times\sqrt{-1}\times\sqrt{x^{6}}\times\sqrt{x}=6\sqrt{3}x^{3}i\sqrt{x}\)

Step19: Solve the quadratic equation \(3x^{2}+3 = 0\)

First, subtract 3 from both sides: \(3x^{2}=-3\), then \(x^{2}=-1\), so \(x=\pm i\)

Step20: Solve the quadratic equation \(5x^{2}+35 = 0\)

Subtract 35 from both sides: \(5x^{2}=-35\), then \(x^{2}=-7\), so \(x=\pm\sqrt{7}i\)

Step21: Solve the quadratic equation \(4x^{2}+120 = 0\)

Subtract 120 from both sides: \(4x^{2}=-120\), then \(x^{2}=-30\), so \(x=\pm\sqrt{30}i\)

Step22: Solve the quadratic equation \(x^{2}-16 = 0\)

Add 16 to both sides: \(x^{2}=16\), so \(x=\pm4\)

Step23: Solve the quadratic equation \(8x^{2}+96 = 0\)

Subtract 96 from both sides: \(8x^{2}=-96\), then \(x^{2}=-12\), so \(x=\pm2\sqrt{3}i\)

Step24: Solve the quadratic equation \(\frac{3}{4}x^{2}+12 = 0\)

Subtract 12 from both sides: \(\frac{3}{4}x^{2}=-12\), then \(x^{2}=-16\), so \(x=\pm4i\)

Answer:

1. \(2 - i\)
2. \(2 + i\)
3. \(7 + i\)
4. \(18-13i\)
5. \(20\)
6. \(-60i\)
7. \(-294i\)
8. \(23-14i\)
9. \(\frac{1}{5}+\frac{2}{5}i\)
10. \(\frac{14}{113}+\frac{16}{113}i\)
11. \(\frac{7}{5}+\frac{1}{5}i\)
12. \(-2-\frac{8}{3}i\)
13. \(-i\)
14. \(i\)
15. \(-1\)
16. \(6i\)
17. \(-16\)
18. \(6\sqrt{3}x^{3}i\sqrt{x}\)
19. \(x=\pm i\)
20. \(x=\pm\sqrt{7}i\)
21. \(x=\pm\sqrt{30}i\)
22. \(x=\pm4\)
23. \(x=\pm2\sqrt{3}i\)
24. \(x=\pm4i\)