3.2 simplifying expressions with rational exponents and radicals answer key

Video transcript

- [Voiceover] So I have an interesting equation here. It says V to the negative six fifths power times the fifth root of V is equal to V to the K power, for V being greater than/equal to zero. And what I wanna do is try to figure out what K needs to be. So what is... what is K going to be equal to? So pause the video and see if you can figure out K, and I'll give you a hint, you just have to leverage some of your exponent properties. Alright, let's work this out together. So the first thing I'd want to do is being a little bit consistent in how I write my exponents. So here I've written it as negative six fifths power, and here I've written it as a fifth root, but we know that the fifth root of something... we know that the fifth root... the fifth root of V, that's the same way, that's the same thing as saying V to the one fifth power, and the reason I want to say that is because then I'm multiplying two different powers of the same base, two different powers of V. And so we can use our exponent properties there. So, this is gonna be the same thing as V to the negative six fifths times, instead of saying the fifth root of V I can say times V to the one fifth power is going to be equal to V to the K. It's gonna be equal to V to the K power. Now, if I'm multiplying V to some power times V to some other power, we know what the exponent properties would tell us, and I could remind us. I'll do it over here. If I have X to the A times X to the B, that's going to be X to the A plus B power. So here, I have the same base, V. So this is going to be V to the, and I could just add the exponents. V to the negative six fifths power plus one fifth power, or V to the negative six fifths plus one fifth power is going to be equal to V to the K. Is equal to V to the K. I think you might see where this is all going now. So this is going to be equal to V. So negative six fifths plus one fifth is going to be negative five fifths or negative one. So all of this is going to be equal to negative one, and that's going to be equal to V to the K. So K must be equal to negative one, and we're done. K is equal to negative one.

When simplifying radicals that use fractional exponents, the numerator on the exponent is divided by the denominator. All radicals can be shown as having an equivalent fractional exponent. For example:

Radicals having some exponent value inside the radical can also be written as a fractional exponent. For example:

The general form that radicals having exponents take is:

Should the reciprocal of a radical having an exponent, it would look as follows:

In both cases shown above, the power of the radical is

and the root of the radical is

. These are the two forms that a radical having an exponent is commonly written in. It is convenient to work with a radical containing an exponent in one of these two forms.

Evaluate

.

Converting to a radical form:

First, the cube root of 27 will reduce to 3, which leaves:

Once the radical having an exponent is converted into a pure fractional exponent, then the following rules can be used.

Properties of Exponents

Simplify

.

First, you need to separate the different variables:

becomes

Combining the exponents yields:

Which results in:

Which simplifies to:

Simplify

.

First, separate the different variables:

becomes

[1]

Combining the exponents yields:

Which gives:

Which simplifies to:

Questions

Write each of the following fractional exponents in radical form.

1. 
2. 
3. 
4. 
5. 
6. 

Write each of the following radicals in exponential form.

1. 
2. 
3. 
4. 
5. 
6. 

Evaluate the following.

1. 
2. 
3. 
4. 

Simplify. Your answer should only contain positive exponents.

1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 

Answer Key 9.6

---