Video: Simplifying a Complex Number in Exponential Form

Express the complex number 𝑧 = 𝑒^(βˆ’4 βˆ’ (23πœ‹/12)𝑖) in exponential form.

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Video Transcript

Express the complex number 𝑧 equals 𝑒 to the power of negative four minus 23πœ‹ over 12𝑖 in exponential form.

Now, you might be looking at this question thinking, well 𝑧 is already in exponential form. And at first glance, it does indeed look like it might be. However, we recall that a complex number written in an exponential form is 𝑧 equals π‘Ÿπ‘’ to the π‘–πœƒ. 𝑖 is known as the magnitude or modulus of the complex number, whereas πœƒ is its argument. And what’s important here is that we’re going to define πœƒ in terms of its principal value. That’s value of πœƒ greater than negative πœ‹ and less than or equal to πœ‹. So we go back to our complex number and see what we can do to write it in this form.

Well, let’s begin by recalling that π‘₯ to the power of π‘Ž times π‘₯ to the power of 𝑏 is π‘₯ to the power of π‘Ž plus 𝑏. This means we can rewrite 𝑒 to the power of negative four minus 23πœ‹ over 12𝑖 as 𝑒 to the power of negative four times 𝑒 to the power of negative 23πœ‹ over 12𝑖. And we have our value for π‘Ÿ. It’s 𝑒 to the power of negative four πœƒ is negative 23πœ‹ over 12. Remember, we want our principal value to be greater than negative πœ‹ and less than or equal to πœ‹. This is clearly outside of this range.

So we recall that we can find the principal value by adding or subtracting multiples of two πœ‹ to the value for πœƒ. So we get negative 23πœ‹ over 12 plus two πœ‹. And since we can write two πœ‹ as 24πœ‹ over 12, we find that the principal value of πœƒ must be πœ‹ by 12. And we’ve expressed our complex number in exponential form. 𝑧 is equal to 𝑒 to the power of negative four times 𝑒 to the power of πœ‹ over 12𝑖.

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