Simplify each power of i. i^-14

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 -98th power. So in this case you can see that we have a component to the power to a negative power. So recall that we can write a variable that has a negative power as a fraction. So say I have a to the negative end. I can rewrite it as one divided by a to the end. So see now that I've written the exponents and the denominator of the fraction the end is now positive. So we can do that too. Our expression. So I to the -98 can be rewritten as one divided by I to the positive 98. And now that we have a positive exponent we can focus on simplifying I to the power of 98. But let's think about what I represent. So I is the imaginary number. The square root of negative one. So if I have I squared that is equal to the square root of negative one times the square root of negative one. Or just -1. So I want to try to pull as many I to the second power from I to the 98th power because I know that I to the second power is negative one. So if I have I squared I don't know how many I'm pulling out but I know that when I multiply two times X. I need to get my original power back which is 98. So if I sell for X and see that X is equal to 49. So that means that there's 49 I squared in I to the 98. And I know that I to the second power is negative one so I can plug that in and now I have negative one to the power of 49. So because we're raising a negative number to an odd power, I'm still going to get back any negative number. So I'm going to be getting back negative one. You can think about this because of when I have negative one squared, that is equal to negative one times negative one. And that gives me back positive one. But if I have negative one to the third power that gives me negative one times negative one, times negative one. And that still gives me back A negative number, -1. So because 49 is an odd power, I'm still going to get back negative one. Had it been 48 or 50, I would have gotten back positive one. But in this case I get back negative one. But recall that my expression one divided by eye to the 98 means that I to the 98 is in the denominator but I need to plug in negative one here. So this becomes one divided by negative one, Which simplifies to just be negative one. So that means that the expression I to the -98 is equal to negative one. And if we look at the answer choices, you can see that A is also negative one. So hopefully this video helped and see you next time.

Simplify each power of i. i^-14

Key Concepts

Imaginary Unit i

Powers of i and Their Cyclic Pattern

Negative Exponents

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Simplify each power of i. i-14

Key Concepts

Imaginary Unit i

Powers of i and Their Cyclic Pattern

Negative Exponents

Watch next

Simplify each power of i. i^-14