In Exercises 59–76, find the indicated root, or state that the ex...

Question

In Exercises 59–76, find the indicated root, or state that the expression is not a real number.___⁴√−16

Accepted Answer

Hey, everyone in this problem, we're asked to determine the indicated root in the given expression. The expression we're given is the 4th root of -625. We have four answer choices. Option A is 25. Option B is -5. Option C is five and option D is not a real number. Let's start by rewriting what we were given. So we have the fourth route of -625. Now notice that we're taking the even route of a negative numbers. The even route of a negative number always gives us not a real number. So whenever you have a problem like this inside of that route, you have a negative number and you have an even route, you're gonna get not a real number. Now, if you don't remember this rule, what you can do is break this route up into something related to a square root. OK. We know when we have the square root of a negative number that that's not a real number. All right. So recall, we can write the 4th route of -625 as - to the exponent one divided by four. We have a rule that allows us to relate the root, the radical to an exponent. OK. Now also remember the rule where if we have say A to the exponent M multiplied by N, we can break this up and write it as a to the exponent M to the exponent end. So in our case, we have an exponent of 1/4, we want to relate this to the square root. We know the square root is represented through an exponent of one half. So if we can relate one quarter to one half, we will be able to relate this to the square root. So 1 over 4 recognize that it can be written as 1/2 multiplied by 1/2. So we can write negative 625 to the exponent 1 over As -625 to the exponent 1/2 to the exponent 1/2. OK? When we multiply these exponents together one half multiplied by one half, it gives us our original one divided by four, no, inside these brackets, we have negative 625 to the exponent one half. We know that that is equivalent to the square root Of -625. And all of this is raised to the exponent one half. Well, we know that negative of a square root, sorry, the square root of a negative number is going to give us not a real number. OK? So if we have not a real number and then we take the square root again, it's still going to be not a real number. So it's really, really handy to remember this rule. If you have an even root of a negative number, you're gonna get not a real number. But if you don't remember that rule and you're in a problem, you can work it out. Try to relate it to the square root. OK? Because you know what happens when we have the square root of a negative number and that's gonna show you that you get not a real number. OK? That corresponds with answer choice. D Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 59–76, find the indicated root, or state that the expression is not a real number.___⁴√−16

Key Concepts

Even Roots of Negative Numbers

Complex Numbers

Radical Expressions

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