The expression \( 6[e^{i \cdot 270^\circ}] \) involves the complex number in polar form.

In polar form, \( e^{i \theta} \) represents a complex number on the unit circle with an angle \( \theta \) measured counterclockwise from the positive real axis.

1. The angle \( 270^\circ \) is equivalent to \( \frac{3\pi}{2} \) radians.
2. Using Euler's formula, we know \( e^{i \theta} = \cos \theta + i \sin \theta \).
   
For \( \theta = 270^\circ \) or \( \frac{3\pi}{2} \) radians:

\[
e^{i \cdot 270^\circ} = \cos(270^\circ) + i \sin(270^\circ)
\]
\[
e^{i \cdot 270^\circ} = 0 - i = -i
\]

Now, multiplying by 6:

\[
6 \cdot e^{i \cdot 270^\circ} = 6 \cdot (-i) = -6i
\]

So, the result is \( -6i \).

Would you like more details or have any questions about this solution?

Here are 5 related questions:
1. How is Euler's formula derived?
2. What is the polar form of a complex number?
3. Can you convert \( 6[e^{i \cdot 180^\circ}] \) to rectangular form?
4. How does multiplying a complex number by a scalar affect its magnitude and direction?
5. What is the difference between degrees and radians in angular measurement?

**Tip:** When working with complex numbers, the angle in polar form is usually expressed in radians for most calculations.

The query submitted by the user: 6[ei⋅270°]

Learn how to solve the complex number 6[e^{i⋅270°}] using Euler's formula. This problem involves converting polar form to rectangular form and multiplying complex numbers. Suitable for high school students in Grades 10-12.

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