Question Video: Simplifying Imaginary Numbers Mathematics

Video Transcript

Simplify one divided by 𝑖 raised to the power of negative 39.

In this question, we’re asked to simplify an expression involving 𝑖 raised to an integer exponent. And whenever we’re asked to simplify an expression like this, the easiest way is to recall that 𝑖 to the fourth power is equal to one. And we know this is true because 𝑖 is the square root of negative one, so 𝑖 squared will be equal to negative one. And then 𝑖 to the fourth power must be negative one squared; it must be equal to one. So to use this to simplify our expression, we’re first going to need to write our expression in terms of 𝑖 to the fourth power. To do this, we notice we have a negative integer exponent in our denominator, and we can simplify this. One divided by 𝑎 to the power of negative 𝑛 is just equal to 𝑎 to the 𝑛th power.

In our case, our value of 𝑎 is 𝑖 and 𝑛 is 39. So this gives us that one over 𝑖 raised to the power of negative 39 is just equal to 𝑖 to the 39th power. We want to write this expression in terms of 𝑖 to the fourth power. And there’s a few different ways we could do this. We could write this product out in full. However, the easiest way is probably to notice that 39 is equal to four times nine plus three. In other words, if we were to write this product out in full, we would have nine factors of 𝑖 to the fourth power and we would have three factors of 𝑖 left over. And we can then use this to simplify our expression. First, we’ll split our exponent, so this is equal to 𝑖 to the power of four times nine multiplied by 𝑖 cubed. Next, 𝑖 to the power of four times nine is 𝑖 to the 36 power. This is exactly the same as 𝑖 to the fourth power all raised to the ninth power.

We can see this either by using our laws of exponents or by writing this product out in full. Finally, we just use two facts about 𝑖. First, 𝑖 to the fourth power is equal to one. So 𝑖 to the fourth power all raised to the ninth power is just equal to one to the ninth power. And of course, one raised to the ninth power is just equal to one. Secondly, we’re going to use the fact that 𝑖 is the square root of negative one. So 𝑖 squared is equal to negative one. This means that 𝑖 cubed is 𝑖 squared times 𝑖, which is negative 𝑖. So we simplify this expression to get one multiplied by negative 𝑖, which is of course just equal to negative 𝑖.

Therefore, by using the fact that 𝑖 squared is negative one and 𝑖 to the fourth power is equal to one, we were able to simplify one divided by 𝑖 to the negative 39th power to be negative 𝑖.