(2+3i)^6
Using Demoirves Theorem.
I tried solving it out, but somehow it didn't work for me.
(2+3i)^6
r=|z|=square root of 2^2 + 3^2 = square root of 13
z = square root of 13 *
(cos____ + isin_______ )
At this point I am stuck, so can someone help me?
let z = 2 + 3i
for the angle, tanØ = 3/2
Ø = .98279 radians
so z = √13(cos .98279 + isin .98279)
then z^6
= √3^6(cos 6(.98279) + isin 6(.98279) )
= 27(cos 5.89676 + i sin 5.89676)
= appr 25 - 10.18 i
or in short form:
27 cis 5.89676
If you need the answer is terms of degrees, set your calculator to degrees and repeat my steps
Thank you so much Reiny! I finally understand it!
somehow the √13 became √3, but I'm sure David can fix that, eh?
To solve the expression (2 + 3i)^6 using De Moivre's Theorem, follow these steps:
Step 1: Find the magnitude of the complex number.
The magnitude (r) is given by the square root of the sum of the squares of the real and imaginary parts:
r = √(2^2 + 3^2) = √13.
Step 2: Write the complex number in polar form.
To convert the complex number (2 + 3i) into polar form, you need to determine the angle (θ) using the inverse tangent function:
θ = tan^(-1)(3/2).
Step 3: Apply De Moivre's Theorem.
De Moivre's Theorem states that for any non-negative integer n and any complex number z = r(cosθ + isinθ), the value of z^n is given by:
z^n = r^n(cos(nθ) + isin(nθ)).
So in this case, we have:
(2 + 3i)^6 = (√13)^6 * [cos(6θ) + isin(6θ)].
Step 4: Evaluate the angles.
To find the angle for the sixth power, multiply the original angle by 6:
θ = 6 * tan^(-1)(3/2).
Step 5: Substitute the values into the formula.
(2 + 3i)^6 = (√13)^6 * [cos(6θ) + isin(6θ)] = 13^3 * [cos(6θ) + isin(6θ)].
Step 6: Simplify.
Evaluate cos(6θ) and sin(6θ) using the value of θ obtained in Step 4.
Step 7: Calculate the real and imaginary parts of the final answer.
Multiply 13^3 by cos(6θ) to get the real part.
Multiply 13^3 by sin(6θ) to get the imaginary part.
By following these steps, you should be able to simplify and calculate the final answer. Remember to use a calculator to evaluate trigonometric functions accurately.