Question Video: Simplifying Imaginary Numbers Mathematics

Video Transcript

Simplify 𝑖 to the power of negative 54.

In this question, we’re asked to simplify 𝑖 raised to an integer exponent. So the easiest way to answer this question will be to use our theorem involving raising 𝑖 to integer exponents. But this isn’t the only way we can do this. We’re going to do this in full because it can often be useful to see the process written out in full rather than just applying a theorem.

First, recall whenever we raise a number to a negative exponent, we can instead divide by that number raised to the positive exponent. 𝑖 to the power of negative 54 is equal to one over 𝑖 raised to the 54th power. But we know exactly what 𝑖 to the 54th power means. We know whenever we raise a number to a positive integer exponent, for example, 54, then this is the same as the product of that number with itself where it appears 54 times in our product. So 𝑖 to the 54th power is 𝑖 times 𝑖 times 𝑖, where 𝑖 appears 54 times in this product.

We have a few choices of how we could simplify this. We’re just going to use the fact that because 𝑖 is the square root of negative one, 𝑖 squared must be equal to negative one. So we want to write our product in the denominator in terms of 𝑖 squared. So we pair off each factor of 𝑖 with another factor of 𝑖 to get factors of 𝑖 squared. But if 𝑖 appears 54 times in this product, then 𝑖 squared will appear 54 over two times. It will appear 27 times in this product.

Now, we can just replace each factor of 𝑖 squared in our denominator with negative one. Now we have two choices; either we could write the denominator as negative one all raised to the 27th power, or we could just evaluate this expression directly. We know negative one multiplied by negative one is equal to one. So if we multiply negative one by itself 27 times, then 26 of these times will cancel to give us one. So we just get one over negative one. And one divided by negative one is equal to negative one. Therefore, we were able to show that 𝑖 to the power of negative 54 is just equal to negative one.