Evaluate (5+5√3i)^7 using DeMoivre’s theorem.
Write your answer in rectangular form. Show all your work.
let z = 5+5√3i = 5(1 + √3i)
magnitude = 5√(1+3) = 10
tanØ = 5/√3/5 = √3
Ø = 60°
z = 10(cos60° + i sin60°)
z^7 = 10^7(cos420° + i sin 420°)
= 10^7(cos60 + isin 60) , since cos 420 = cos(360+60) = cos60 , same for sin420
= 10^7/2 + i√3 10^7/2
To evaluate (5 + 5√3i)^7 using DeMoivre's theorem, we first need to express the complex number in polar form.
Step 1: Convert to polar form
Let's find the magnitude (r) and argument (θ) of the complex number (5 + 5√3i).
The magnitude (r) is given by:
r = √(a^2 + b^2)
r = √(5^2 + (5√3)^2)
r = √(25 + 75)
r = √100
r = 10
The argument (θ) is given by:
θ = arctan(b/a)
θ = arctan((5√3) / 5)
θ = arctan(√3)
θ ≈ 1.047
So, (5 + 5√3i) in polar form is (r, θ) = (10, 1.047).
Step 2: Apply DeMoivre's theorem
DeMoivre's theorem states that for any complex number in polar form (r, θ) raised to the power of n, the result is:
(r, θ)^n = (r^n, nθ)
Applying DeMoivre's theorem to our complex number (10, 1.047)^7, we get:
(10^7, 7 * 1.047)
Step 3: Evaluate the polar form
Calculating the magnitude:
10^7 = 10,000,000
Calculating 7 * 1.047:
7 * 1.047 ≈ 7.329
So, the polar form of (5 + 5√3i)^7 is (10, 7.329).
Step 4: Convert back to rectangular form
To convert back to rectangular form, we use the polar coordinates (r, θ) and the trigonometric relations:
x = r * cos(θ)
y = r * sin(θ)
Calculating the real part (x):
x = 10 * cos(7.329)
x ≈ 10 * 0.754
x ≈ 7.54
Calculating the imaginary part (y):
y = 10 * sin(7.329)
y ≈ 10 * 0.657
y ≈ 6.57
So, the rectangular form of (5 + 5√3i)^7 is approximately 7.54 + 6.57i.
To evaluate the expression (5+5√3i)^7 using DeMoivre's theorem, we need to convert the complex number into polar form and then apply the theorem.
First, let's convert the complex number 5+5√3i into polar form. The polar form of a complex number is given by r cis(θ), where r is the magnitude of the complex number and θ is the argument of the complex number.
The magnitude of the complex number can be found using the formula:
|r| = √(a^2 + b^2)
In our case, a = 5 and b = 5√3. Therefore, the magnitude is:
|r| = √(5^2 + (5√3)^2) = √(25 + 75) = √100 = 10
To find the argument (θ), we use the formula:
θ = arctan(b/a)
In our case, a = 5 and b = 5√3. Therefore, the argument is:
θ = arctan((5√3)/5) = arctan(√3) = π/3
Now, we have the complex number 5+5√3i in polar form: 10 cis(π/3).
Next, we can raise this number to the power of 7 using DeMoivre's theorem, which states that for any complex number r cis(θ) raised to the power of n, the result is r^n cis(nθ).
So, applying DeMoivre's theorem, we have:
(10 cis(π/3))^7 = 10^7 cis(7(π/3))
Now, let's calculate the values:
10^7 = 10000000
7(π/3) = (7/3)π
Therefore, the expression simplifies to:
(10 cis(π/3))^7 = 10000000 cis((7/3)π)
Finally, let's convert this back into rectangular form using Euler's formula:
10000000 cis((7/3)π) = 10000000(cos((7/3)π) + i sin((7/3)π))
Now, let's evaluate cos((7/3)π) and sin((7/3)π):
cos((7/3)π) = -1/2
sin((7/3)π) = √3/2
Substituting these values back:
10000000(cos((7/3)π) + i sin((7/3)π)) = 10000000(-1/2 + i√3/2)
Simplifying further:
= -5000000 + 5000000√3i
Therefore, the expression (5+5√3i)^7 in rectangular form is -5000000 + 5000000√3i.