Express the complex number 𝑧 equals 𝑒 to the power of negative four minus 23𝜋 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 𝑟𝑒 to the 𝑖𝜃. 𝑖 is known as the magnitude or modulus of the complex number, whereas 𝜃 is its argument. And what’s important here is that we’re going to define 𝜃 in terms of its principal value. That’s value of 𝜃 greater than negative 𝜋 and less than or equal to 𝜋. 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 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. This means we can rewrite 𝑒 to the power of negative four minus 23𝜋 over 12𝑖 as 𝑒 to the power of negative four times 𝑒 to the power of negative 23𝜋 over 12𝑖. And we have our value for 𝑟. It’s 𝑒 to the power of negative four 𝜃 is negative 23𝜋 over 12. Remember, we want our principal value to be greater than negative 𝜋 and less than or equal to 𝜋. 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 to the value for 𝜃. So we get negative 23𝜋 over 12 plus two 𝜋. And since we can write two 𝜋 as 24𝜋 over 12, we find that the principal value of 𝜃 must be 𝜋 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 over 12𝑖.