Question Video: Simplifying a Complex Number in Exponential Form Mathematics

Video Transcript

Express the complex number 𝑧 is equal to 𝑒 to the power of negative four minus 23𝜋 over 12 𝑖 in the form of 𝑟 multiplied by 𝑒 to the power of 𝜃𝑖.

In this question, we’re asked to express the complex number in exponential form such that 𝑧 is equal to 𝑟 multiplied by 𝑒 to the power of 𝜃𝑖, where 𝑟 is the magnitude, or modulus, of the complex number and 𝜃 is its argument.

We will begin by defining 𝜃 in terms of its principal value, that is, the value of 𝜃 that is greater than negative 𝜋 and less than or equal to 𝜋. Let’s go back to our complex number and see how we can write it in this form. We recall from our laws of exponents or indices that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. This means that we can rewrite the complex number as 𝑒 to the power of negative four multiplied by 𝑒 to the power of negative 23𝜋 over 12 𝑖.

We now have our value for 𝑟. It’s 𝑒 to the power of negative four. 𝜃 is negative 23𝜋 over 12. However, we want our principal value to be greater than negative 𝜋 and less than or equal to 𝜋. And clearly, our value is outside of this range. We recall that we can find the principal value by adding or subtracting multiples of two 𝜋 to the value of 𝜃. In this case, we will add two 𝜋 to negative 23𝜋 over 12. We can write two 𝜋 as 24𝜋 over 12, giving us negative 23𝜋 over 12 plus 24𝜋 over 12. This is equal to 𝜋 over 12. And we can therefore express our complex number as 𝑒 to the power of negative four multiplied by 𝑒 to the power of 𝜋 over 12 𝑖. This is the exponential form of the complex number 𝑒 to the power of negative four minus 23𝜋 over 12 𝑖.