Simplify each power of i. i^-13

Question

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we want to simplify I to the -173. So in this case we're working with I which is a complex number that represents the square root of negative one or the imaginary number. So in order to simplify let's think about what I means if we square it so I squared is equal to the square root of negative one times the square root of negative one which is actually equal to -1. And notice how -1 is now a real number and no longer imaginary. So this will help us simplify or expression I to the negative because I want to try to pull out as many real numbers as I can which means I want to find how many I squared. I can pull out from 173. Well before I can start doing that I have to deal with this negative. So recall that when an expression has a negative exponents say A to the negative end, I can rewrite that as a fraction to make the exponent positive. So A to the negative end becomes one divided by a to the end and notice how when the exponent is in the denominator it is now positive. So I'm going to express I to the - as one over I to the positive 173. Now I can focus on simplifying or pulling out as many I Squared from I to the 173. I just need to be aware that it's going to be in the denominator of a fraction. So in this case 173 is a odd number. So I need to split it up in order to get an even number that I can pull as many I squares out from. So I'm going to rewrite this as one divided by I to the 172 plus one, which I can then rewrite as One divided by eye to the 172 times I to the first power. So now I have 172 and if I Divide that by two I get So that means that I can pull out 86 I squared from I to the 172. And recall that I squared is negative one. So now I have negative one to the 86th power and notice how this is an even power. So that means that the end result will be positive because recall that a negative number raised to an even power gets me a positive answer. So say I have negative one times negative one. That is the same as negative one squared, which gives me back positive one. So that same principle applies to negative one to the power of 86. So I end up getting back one and now I can plug this back into I To the 172 because that is what we simplified. So if I plugged that in, I have one divided by one times I to the first power. So this simplifies to one over I to the first power. And if I want to get rid of the I in the denominator, we'll go ahead and multiply by I over I. So this becomes I over I squared and recall that I squared is equal to -1. So this is I over negative one, which I can then write as negative I So negative I becomes my final simplification for I to the -173 power. And you can see that negative. I also aligns with answer choice C. So hopefully this video helped and see you next time.

Simplify each power of i. i^-13

Key Concepts

Imaginary Unit i and its Powers

Negative Exponents

Modular Arithmetic for Exponent Reduction

Watch next

Simplify each power of i. i-13

Key Concepts

Imaginary Unit i and its Powers

Negative Exponents

Modular Arithmetic for Exponent Reduction

Watch next

Simplify each power of i. i^-13