Iknow that these answers are correct, but i don't understand how to get it using the de moivere theorem. can someone explain in detail? you so much!

Step-by-step explanation:

10.

First, convert 1+i from Cartesian to polar.

r = √(1² + 1²)

r = √2

θ = atan(1/1), θ in first quadrant

θ = 45°

Therefore:

(1+i)²⁰ = (√2 (cos 45° + i sin 45°))²⁰

(1+i)²⁰ = 1024 (cos 45° + i sin 45°)²⁰

Now applying the De Moivre theorem:

1024 (cos (20×45°) + i sin (20×45°))

1024 (cos (900°) + i sin (900°))

1024 (-1 + 0)

-1024

11.

Repeat the same steps from Question 10.  First, convert to polar:

r = √(1² + (-1)²)

r = √2

θ = atan(-1/1), θ in fourth quadrant

θ = 315°

Therefore:

(1−i)¹⁰ = (√2 (cos 315° + i sin 315°))¹⁰

(1−i)¹⁰ = 32 (cos 315° + i sin 315°)¹⁰

Now applying the De Moivre theorem:

32 (cos (10×315°) + i sin (10×315°))

32 (cos (3150°) + i sin (3150°))

32 (0 − i)

-32i

step-by-step explanation:

the original amount doesn't ''re just looking for the time it takes for 1/2 of the substance to we have

(1/2) =)     take the ln   of both sides

ln (1/2)   = )   and we can write

ln(1/2) = ) lne     and lne   = 1     so we have

ln(1/2) = -.1092t     divide both sides by -.1092

t = ln(1/2)/-.1092 = about 6.347 days = about 6.3 days (rounded)

step-by-step explanation:

first, eliminate the parenthesis:

3

x

+

2

−

4

x

second, group like terms:

3

x

−

4

x

+

2

third, combine like terms:

(

3

−

4

)

x

+

2

→

−

1

x

+

2

→

−

x

+

2