Write the complex number -4 x 3^1/3 - 4i in exponential form.
I'm not sure how I use euler's formula to solve for this. A brief explanation would be greatly appreciated.
Does the "x" indicate multiplication, or is it an unknown? Does the 4 multi-ply (3^1/3) - 4i or just 3^(1/3) ?
Find the magnitude of the complex number first. Call it C. Then write your number as
C*e^(i*z)
using Euler's formula.
e^iz = cos z + i sin z
The magnitude of
-4 *(3^1/3 - 4i)
is
4*sqrt[3^(2/3) + 16] = 17.01
so you can write
-4 *(3^1/3 - 4i)
= 17.01*(-0.3392 +0.9406 i)
Now find the number z (in radians) for which
cos z = -0.3392 and sinz = 0.9406
It will be in the second quadrant
z = 1.9169
The answer would be 17.01 exp(1.9169i),
but it depends upon whether I interpreted what you wrote correctly. You need to use parenthese and explain the "x"
Sorry for the misunderstanding. The X is a multiplication sign and the 4 is multiplied to the cubed root of 3.
To write the complex number -4 × 3^(1/3) - 4i in exponential form, we can apply Euler's formula.
Euler's formula is given by:
e^(iθ) = cos(θ) + i sin(θ)
First, let's find the angle θ in our complex number. Bear in mind that the argument of a complex number - in this case, the angle - can be found using the formula:
θ = arctan(b/a)
where a is the real part of the complex number and b is the imaginary part.
In this case, the complex number is -4 × 3^(1/3) - 4i. Therefore, a = -4 × 3^(1/3) and b = -4.
Using these values, we can calculate θ as follows:
θ = arctan((-4)/(-4 × 3^(1/3)))
= arctan(1/3^(1/3))
≈ arctan(0.693)
≈ 35.1°
Now that we know θ, we can write the complex number in exponential form using Euler's formula:
z = |z| × e^(iθ)
where |z| is the magnitude (absolute value) of the complex number.
The magnitude of the complex number is calculated using the modulus formula:
|z| = sqrt(a^2 + b^2)
In our case, a = -4 × 3^(1/3) and b = -4. Substituting these values, we find:
|z| = sqrt[(-4 × 3^(1/3))^2 + (-4)^2]
= sqrt[16 × 3^(2/3) + 16]
= sqrt[16(3^(2/3) + 1)]
= sqrt[16 × 3^(2/3) + 16]
Therefore, the complex number -4 × 3^(1/3) - 4i in exponential form is:
z = sqrt[16 × 3^(2/3) + 16] × e^(i35.1°)