01 - Simplify Rational Exponents (Fractional Exponents, Powers & Radicals) - Part 1

Summary

This algebra lesson introduces rational exponents, also known as fractional exponents, as a stepping stone to understanding exponential functions and logarithms. The instructor explains how rational exponents relate to roots, demonstrating that a 1/2 power is a square root, 1/3 is a cube root, and so on. The lesson explores complex exponents like 2/3, showing that they can be calculated by either squaring first then taking the cube root or vice versa. The instructor emphasizes that the order of operations doesn't affect the result due to the properties of exponents, and provides examples to solidify the concept.

Takeaways

• 📚 The lesson focuses on rational exponents, which are exponents containing fractions.
• 🔢 Basic exponent rules are reviewed, such as squaring a number (5^2 = 25) and negative exponents (5^-2 = 1/(5^2)).
• 🆚 Negative exponents are handled by inverting the base and making the exponent positive.
• 🎯 Any number to the zero power is defined as 1, a fundamental rule in exponents.
• 🛑 The lesson transitions into more complex problems to prepare for studying exponential functions and logarithms.
• 🔄 Rational exponents are introduced as fractions, where the numerator indicates the power and the denominator the root.
• 📐 Examples are given to show that B^(1/2) is the square root of B, and B^(1/3) is the cube root of B.
• 🔄 The concept is expanded to show that (B^(P/Q)) is equivalent to (B^P)^(1/Q) or (B^(1/Q))^P, demonstrating the order of operations in exponents.
• 🧮 Practical examples are solved to illustrate the process of calculating expressions with rational exponents, such as 81^(1/2) which equals 9.
• ✅ The lesson emphasizes understanding the rules of exponents rather than memorizing formulas, especially the multiplication of exponents when raising a power to another power.

Q & A

What are rational exponents?

Rational exponents are another way of saying fractional exponents, where the exponent contains a fraction. They are exponents that have a fraction in the denominator, indicating the root to be taken, and a numerator indicating the power to be applied after taking the root.

How is a negative exponent different from a positive one?

A negative exponent is the reciprocal of a positive exponent. For example, if you have a number 'a' raised to the power of -n, it is equivalent to 1/(a^n), where 'n' is a positive integer.

Why is any number raised to the power of 0 equal to 1?

In mathematics, any non-zero number raised to the power of 0 is defined as 1. This is a fundamental property of exponents and is used as a convention to simplify expressions and maintain consistency in mathematical operations.

Can you provide an example of converting a fractional exponent to a radical?

Yes, for instance, if you have a number 'b' raised to the power of 1/2, it is the same as the square root of 'b'. So, b^(1/2) is equivalent to the square root of 'b'.

What is the significance of the numerator and denominator in a rational exponent?

In a rational exponent like 'a^(p/q)', the denominator 'q' indicates the root to be taken (e.g., 2 for square root, 3 for cube root), and the numerator 'p' indicates the power to which the root is raised.

How do you simplify an expression with a rational exponent like 2/3?

To simplify an expression with a rational exponent like 2/3, you can either raise the base to the power of the numerator and then take the root indicated by the denominator, or take the root first and then raise the result to the power of the numerator. The order does not matter due to the properties of exponents.

What happens when you have a negative base and a rational exponent?

When you have a negative base and a rational exponent, you must consider whether the root is even or odd. For even roots (like square roots), the result is an imaginary number. For odd roots (like cube roots), the result is a real number, just like with positive bases.

Can you give an example of a complex rational exponent problem?

Sure, an example of a complex rational exponent problem could be 27 raised to the power of 2/3. This can be simplified by either squaring the base first and then taking the cube root, or by taking the cube root first and then squaring the result.

Why is it often easier to take the root first when simplifying rational exponents?

It is often easier to take the root first when simplifying rational exponents because it can reduce the size of the numbers you are working with, making the arithmetic simpler and potentially avoiding the need for a calculator for large numbers.

How do you handle negative exponents with rational exponents?

Negative exponents with rational exponents are handled by converting them to positive exponents and then taking the reciprocal of the result. For example, 'a^(-n/m)' becomes '1/(a^(n/m))'.

Outlines

00:00

📚 Introduction to Rational Exponents

The paragraph introduces the concept of rational exponents, also known as fractional exponents, which are exponents that include fractions. It connects this concept to previously learned topics such as radicals and negative exponents. The instructor emphasizes the importance of understanding rational exponents as a foundation for studying exponential functions and logarithms. Basic examples are provided, such as 5 squared equals 25, and negative exponents are explained by converting them to positive exponents with a reciprocal base, like 5 to the negative 2 becoming 1 over 25. The zero exponent rule is also reviewed, where any number raised to the power of zero equals one.

05:00

🔍 Deep Dive into Rational Exponents

This section delves deeper into rational exponents by explaining how they relate to roots. For instance, a number raised to the power of 1/2 is equivalent to the square root of that number. The instructor provides a proof to demonstrate the equivalence of fractional exponents and radicals by squaring both sides of an equation with a fractional exponent, resulting in the original number, thus proving the identity. The paragraph also covers how to handle more complex rational exponents, such as 2/3, by breaking them down into a series of operations involving squaring and taking cube roots.

10:03

🔢 Practical Application of Rational Exponents

The paragraph focuses on applying the knowledge of rational exponents to solve practical problems. It illustrates how to calculate expressions like 81 to the 1/2 power by recognizing it as the square root of 81, which equals 9. The concept is further expanded to include negative rational exponents, such as 49 to the negative 1/2 power, which is the square root of 49 and equals 1/7. The instructor advises on the most efficient approach to solving these problems, often opting to take the root first before raising to a power to simplify calculations.

15:03

🎓 Understanding the Order of Operations with Exponents

This section discusses the importance of the order of operations when dealing with rational exponents. It explains that the order in which you perform the root and the power operations does not affect the final result, due to the properties of exponents. The instructor uses examples to show that whether you square a number first and then take the cube root, or take the cube root first and then square the result, you will arrive at the same answer. This reinforces the idea that the numerator of the rational exponent determines the power, and the denominator determines the root, regardless of the order of operations.

20:04

📉 Negative Exponents and Roots

The paragraph addresses the nuances of handling negative numbers with rational exponents. It clarifies that while the square root of a negative number results in an imaginary number, the cube root of a negative number is a real number and exists. The instructor uses the example of the cube root of -125 to demonstrate this concept. The explanation highlights the difference between even and odd roots when dealing with negative numbers and emphasizes the importance of understanding how exponents apply within parentheses versus outside of them.

25:04

🔄 Correctly Applying Exponents to Negative Numbers

This section clarifies a common mistake regarding the application of rational exponents to negative numbers. It explains that without parentheses grouping a negative number with the exponent, the exponent does not apply to the negative sign. The instructor uses the example of -8 to the 2/3 power, which results in -4, to illustrate this point. The explanation emphasizes the significance of parentheses in determining the correct application of exponents and the order in which operations are performed.

🚀 Conclusion and Future Lessons

The final paragraph summarizes the key takeaways from the lesson on rational exponents. It stresses the importance of understanding that the denominator of a rational exponent indicates the type of root to take, while the numerator indicates the power to apply. The instructor reminds students that the order of performing the root and power operations is flexible, but it's often easier to take the root first. The paragraph concludes by encouraging students to practice solving problems with rational exponents and to continue to the next lesson for further practice.

Keywords

💡Exponential Function

An exponential function is a mathematical function of the form f(x) = a * b^x, where b is any positive number not equal to 1, and a is a constant. In the video, the exponential function is introduced as an important concept that the unit is building up to. The script mentions that understanding rational exponents is a stepping stone to grasping exponential functions, which will be studied in depth later in the unit.

💡Logarithm

A logarithm is the inverse operation to exponentiation, which allows us to solve equations of the form a^b = c for b, where a is the base and c is the result. The video script positions logarithms as a related function to exponentials, indicating that they will be studied after a thorough understanding of rational exponents.

💡Rational Exponents

Rational exponents, also known as fractional exponents, are exponents that include fractions. The script explains that these are a fundamental concept being revisited in depth to prepare for more complex topics like exponential functions and logarithms. An example from the script is B^(1/2), which is explained as the square root of B.

💡Negative Exponents

Negative exponents are those where the power is less than zero. The script clarifies that a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For instance, 5^(-2) is explained as 1/(5^2) or 1/25 in the video.

💡Zero Exponent

Any non-zero number raised to the zero power is defined as 1. This is a basic rule of exponents reiterated in the script to ensure a solid foundation before delving into more complex concepts like rational exponents.

💡Square Root

A square root is the inverse operation of squaring a number, represented by a rational exponent of 1/2. The script uses the square root to explain the concept of rational exponents, stating that B^(1/2) is the square root of B.

💡Cube Root

A cube root is the inverse operation of cubing a number, represented by a rational exponent of 1/3. The script mentions that B^(1/3) is the cube root of B, illustrating how cube roots are a type of rational exponent.

💡Multiplication of Exponents

When an exponent is raised to another exponent, you multiply the exponents. This rule is fundamental to understanding how to handle rational exponents. The script demonstrates this by squaring B^(1/2) to get B^(2/2), which simplifies to B^1 or just B.

💡Order of Operations

The order in which mathematical operations are performed is crucial. The script emphasizes that with rational exponents, you can either take the root first or raise the number to the power first, and the result will be the same due to the properties of exponent multiplication.

💡Decimal Exponents

Decimal exponents are another way to represent rational exponents. The script shows that 0.5 is equivalent to 1/2, and thus can be used to represent square roots. This is used to simplify expressions like 4^(-0.5) to the square root of 4.

💡Imaginary Numbers

Imaginary numbers are a concept briefly touched upon in the script when discussing roots of negative numbers. It is mentioned that while square roots of negative numbers result in imaginary numbers, cube roots (and other odd roots) of negative numbers are real and can be calculated.

Highlights

Introduction to the concept of rational exponents, also known as fractional exponents.

Review of previous knowledge on exponents, including positive and negative exponents.

Explanation that any number raised to the power of 0 equals 1.

Introduction to the idea that fractional exponents are equivalent to roots.

Example of B to the 1/2 power being the same as the square root of B.

Demonstration of how squaring a square root cancels out to the original number.

Proof that fractional exponents and roots are interchangeable through mathematical operations.

Explanation of how to handle more complex rational exponents like 2/3.

Method to rewrite B to the 2/3 power as the cube root of B squared or B to the 1/3 power squared.

General rule for converting any rational exponent B to the P/Q power into a root and a power.

Example calculation of 81 to the 1/2 power as the square root of 81.

Example calculation of 49 to the negative 1/2 power as the reciprocal of the square root of 49.

Step-by-step simplification of 27 to the 2/3 power using both methods of calculation.

Advice on which operation to perform first when dealing with rational exponents to simplify calculations.

Example calculation of 16 to the 3/4 power, emphasizing the order of operations.

Explanation of how to handle negative numbers with fractional exponents, specifically cube roots.

Clarification on the difference between square roots and cube roots of negative numbers.

Example calculation of negative 125 to the negative 1/3 power, showing the cube root of a negative number.

Instruction on converting decimal exponents to fractional form and simplifying.

Final example calculation of negative 8 to the 2/3 power, emphasizing the importance of parentheses.

Summary of the lesson's key takeaways regarding the handling of rational exponents.

Transcripts

hello welcome back to algebra this is

actually a new unit of algebra where the

end game is going to be for us to learn

about the very important concept called

the exponential function and also a

related function called a logarithm

probably most people have heard of the

term exponential and logarithm we're

gonna be culminating this unit of

lessons in studying those extremely

important functions but here in this

lesson we're gonna start off the

discussion by talking about something

called rational exponents and the title

of this lesson

it's called rational exponents another

way to say it is exponents that contain

fractions fractional exponents now I

want to say up in the feet up in the

beginning that we have studied this in

in some degree in the past when we

talked about radicals cubed roots fourth

roots and things like that we talked

about fractional exponents rational

exponents here we're going into much

more depth with much more kind of

complex problems to prepare us to study

the concepts of the exponential function

and logarithms which are coming up very

very soon so we have to review a couple

of things to make sure everybody is on

the same page before we get going so we

we can recall the following things these

are things that you all should know from

previous lessons and we're gonna start

very basic we know that 5 squared that's

an exponent right and we know that

that's 5 times 5 we know that's equal to

25 so far so good not too hard we've

also learned about negative exponents so

here's a positive exponent a negative

exponent might be 5 to the negative 2

and we learned that when we have a

negative exponent all you do is you drop

that guy below a fraction and make the

exponent positive we talked about all

the reasons why this is the case in the

past so if something like this looks

foreign to you you need to go back to

the more basic lessons on negative

exponents when you have a negative

exponent you drop it down make the

exponent positive which means it becomes

1 over 5 times 5 1 over 25 okay and then

we also learned the very important

exponent when we discussed you know

radicals and exponents a long time ago

if we have an exponent any any number

raised to the 0 as an exponent it is by

definition equal to the number 1 now

again if this looks weird to you or if

you've never seen it before go back and

look at the more basic lessons and

exponent we've talked extensively why

raising so

to the zero power actually is defined to

be one in math now what we're doing in

this lesson is we're going a little bit

beyond these basic ideas and we're

talking about rational exponents which

means fractional exponents exponents

that have a fraction and so we might

talk about something like this what if

you have a number I'm gonna represent

that number by a letter B and I'm gonna

raise it to the one-half power this is

what we call a rational exponent because

the exponent has a fraction the

numerator of a fraction is 1 and the

denominator of that fraction is 1/2 so

1/2 there is in the exponent itself now

we've learned this in the past but just

in case you haven't picked it up yet

when you have a fractional exponent like

this let's say 1/2 power the 2 here

means it's going to be a square root so

these two things are interchangeable

right the B to the 1/2 is exactly the

same thing as the square root of B

similarly we have learned in the past

that if you have something like B to the

1/3 power so now there's a 3 on the

bottom instead of a 2 this is going to

be not the square root of B it's gonna

be the cube root of B these are

identical ideas the 1 anything raised to

the 1/3 power actually ends up becoming

a cube root right and then finally just

to kind of give one more example you

might guess what would happen if you

raised B or anything to the 1/4 power

what do you think it would be you see

the pattern here it's the fourth root of

B now we have introduced these concepts

in the past when we talked about

radicals so it shouldn't be completely

foreign to you but again we're going

into a little bit more detail the

question I want to ask you is why is

this the case why is it the case that a

fractional exponent is the same thing as

a root you see the the 1/4 power giving

you a fourth root a one-third power

giving you a cube root and the 1/2 power

this isn't implied 2 because it's a

square root okay why is that the case so

let's take a second just to talk about

why that's the case if this is actually

true if it's true then the following

must also be true we can do anything we

want if these are actually equivalent we

can this is an equation we can do

whatever we want to both sides right so

let's take the B to the one-half power

and let's raise

to the second power we can square the

left-hand side of the equation and then

on the right-hand side of the equation

we'll square it as well so you see all

were doing is squaring both sides of the

equation if this is actually truth and

this is a perfectly valid thing to do

but you know that when you have an

exponent raised to an exponent you just

multiply the exponents together

that's from basic exponent knowledge of

exponents that we've learned a long time

ago two times one half is going to be 2

over 2 because the two times the one and

then the two times the implied 1 this is

a 2 over 1 here and then what do you

have on the right hand side we also know

from our working with radicals that if

you have a square root and you square it

they kind of undo each other and so you

just get a B over here but then you can

see that 2 to the power here this is

just a power of 1 so this is B to the 1

is equal to B and so then B is equal to

B so what we have kind of shown is that

a lot of students say well why is this

true and here's kind of one proof of why

it's true because if I square both sides

of the equation I get exactly the same

thing and so I get the identity that B

is equal to B so that you need to sort

of burn it in your mind that any time

you see a fraction in an exponent it is

the same thing is a radical they're

equivalent there is no difference

between the two it's like saying that I

have ice and I have water

they're both h2o but they're just

slightly different representations of

exactly the same thing when you see a

radical it's exactly the same thing is

an exponent that is a fraction ok so

that's kind of for this now let's just

fly through the other ones here because

you know why not we have a few minutes

what if I take this guy this B to the

1/3 what if I raise him to the third

power then on the right hand side if

this is actually equivalent I would have

to raise him to the third power but I

know that this exponent will be

multiplied by this book exponent which

would be 3 over 3 and I know that a cube

root cancels exactly with a cube we we

all we know from from cubing things we

know that whenever we raise to the power

of the same base of the of the of the

radical there they annihilate each other

and we're left with B and so we end up

with B is equal to B because this is a

first power here and you can imagine the

same exact thing would hold here if I

raise this to the fourth power and raise

this to the fourth power then I'll get B

is equal to B

sort of thing as we have done here

alright so when you see an exponent

that's 1/2 it's a square root if you see

an exponent that's one-third it's a cube

root if you see an exponent that's 1/6

it's a sixth root if you see an exponent

that's 1/10

it's a tenth root I mean you see the

pattern it's not so hard to understand

now we need to go beyond these basic

exponents and talk about what happens if

I have something like what about

something more complicated I told you

we're gonna go a little deeper what

about B to the 2/3 so this is different

because in every example I told you I

said the 1/2 power is a square root the

1/3 power is the cube root the 1/4 power

is the 4th root and so on but this is

not a 1/3 it's 2/3 so that's different

right so what do we have when we have

something like this

well I need you to think about what the

2/3 power really means if I have

something like B to the 2/3 power how

can I write this thing I can write it as

follows I can write it as B squared all

raised to the 1/3 power how do I know I

can do that because remember exponents

whether they're fractional exponents

like these or regular exponents they all

obey the same rules of exponents when

you have a power raised to a power like

this you just multiply the power so we

know that if I multiply 2 times 1/3 I'm

gonna get 2/3 because the 2 times a 1

and then the implied 1 on the bottom

here times 3

I'll get 2/3 back so this is exactly

equivalent of this so I could kind of

break these things apart but then I know

that the 1/3 here is it's a cube root

right so then what I'm saying here is

that the B squared can then be wrapped

up in underneath a cube root because the

B squared is underneath the cube root

applies to the whole thing so what I'm

saying is that B to the 2/3 power can be

written like this but it can also be

written as the cube root of B squared

however I can also write this another

way I can say that B to the 2/3 power

can be written as B to the 1/3 raised to

the 2nd power how do I know I can do

that because

remember exponents you just multiply

them so 2 times 1/3 gives you 2/3 just

like 2 times 1/3 gave me the same 2/3

here all I've done is reverse the order

of what is inside him what is outside so

this is exactly the same thing as this

which is exactly the same thing as this

which is exactly the same thing as this

is kind of like four different ways of

writing the same thing but if I write it

like this then I would take the cube

root first be just be cube root of this

and then whatever that is that whole

entire thing is squared

like this so what I'm basically saying

is that this representation and this

representation is the same thing

literally what I'm saying is there's no

difference at all between this this this

this this and this you see why it gets

complicated a lot of students look at

that in there and you try to memorize

equations and formulas and oh my gosh

I'm gonna try to memorize it no don't

memorize it just understand the

fundamental rules when you have an

exponent raised to another exponent you

just multiply the exponents so anytime I

have a fractional power like 2/3 I can

write it as the squaring coming first

and then the cute the cube root part of

it giving you this or I can write it as

the cube root first and then square it

which gives me something like this the

reason I can do these in any order is

just because multiplying these together

gives me the same thing no matter which

order I do it alright so in your book

you probably will see something like

this in general whatever book you're

using will probably put something like

this if I have B this looks really

confusing in my opinion when you just

read it in a book but now that we've

done this it won't be hard at all if you

have B to the P over Q power that looks

crazy doesn't it what it's basically

saying is that I can write it like this

B to the P and the Q through to that or

I can write it as B to the Q through to

that to the P power now this I admit

looks crazy it looks it looks really

cumbersome and complicated all it's

saying is that if I have a number or a

variable or whatever it is

a fractional power the numerator is P

and the denominator is Q all it's saying

is that I have to take the Q through to

vit because that's on the bottom that's

a sec the cube root or a fourth root or

a square root whatever that is and I

also have to raise B to the power of P

because that's on the top but what it's

saying is it doesn't matter the order in

which I do that I'm gonna get the same

exact thing if I raise it to the power

first and then take the root it's going

to be the same thing is if I take the

root first and then raise it to a power

which is exactly what I showed you here

in terms of a number example if you hide

all of this and I just give you this it

seems really confusing but you can see

that with this

it didn't matter the order and did it

see I squared it first and then I did

the cube root here I did the cube root

first and then I squared it all it's

saying is that when you have a

fractional exponent we call a rational

exponent it doesn't matter if you take

the do the squaring or the power

operation and then the root or the root

and then the power you're gonna get the

same answer if you grab a calculator and

actually do that both different ways you

will get the same number because

mathematically they're the same thing so

this is honestly the entire kind of like

the learning part of this lesson that's

all I really want you to know now what

we have to do is apply what we have kind

of learned here to some actual examples

so we can do that straight away it's not

going to be too bad let's start with

something very very simple what if we

have 81 to the 1/2 power how would we

calculate that or simplify that well the

first thing we recognize is that the 1/2

power is just a square root so this is

the square root of 81 and you all know

that 9 times 9 is 81 or you can write

this as 9 times 9 if you want to and

look for a pair so the answer is 9

circle that is your answer so 81 to the

1/2 power is 9 I encourage you grab a

calculator and actually take 81 and

raise it to the 0.5 power that's 1/2

right and you're gonna find the answer

is at exactly 9 that's what that's what

comes out all right next problem what if

we have 49 raised to the negative 1/2

power how do we simplify that well we

have two things going on we have a

negative

power and it's also a fractional power

so what this means is since it's

negative we're gonna just drop this guy

downstairs and make it a positive

one-half power that's what happens with

negative exponent we drop them down make

them positive but this one half power is

just a square root so this becomes the

square root of 49 square root because

it's a 2 in the bottom of the fraction

and 7 times 7 is 49 you all know that

square root of 49 and so you get 1 over

7 this is the answer all right so far

those are pretty Elementary let's do

something maybe a teeny bit more

challenging what if you have 27 to the

2/3 power this is the first time where

we have this exponent here that's a

fraction but it's a it's not a simple

one like one-third or one-fourth it's

got the 2/3 now we learned just a second

ago that you can do this many different

ways I can do the power first and then I

can cube it a cube root it because

there's got to be a cube root involved

with the 3 in the bottom or I could do

the cube root first and then do the

power later my advice is just pick one

but I'm gonna do it both ways to show

you you know what's happening here let's

say that the first thing we want to do

is squared this this is a 27 and we're

gonna raise that to the power 2 because

there's a 2 here but then we're gonna

wrap the whole thing in parentheses and

raise the result of that to the 1/3

power

if I can write the number 3 correctly

how do I know I can do this because if I

multiply these exponents I'm going to

get 2/3 that's exactly what I started

with that's how you know that this is

legal to do now if you go in a

calculator or grab a sheet of paper and

square the number 27 it comes out to be

a really big number 729 that's big but I

have to take the answer there and raise

it to the 1/3 power right and you all

know that the 1/3 power is just 729 cube

root of that now how do I take the cube

roots you have to take the cube root of

this really big number how do I do that

well you got to grab a calculator or

something to figure out what multiplies

together together and give you 725 when

you play around with it long enough

you're gonna realize that 9 times 81

works and we all know that 81 is 9 times

now now because it's a cube root you're

looking not for pairs you're looking for

triplet

and we found a triplet of nine and so

the answer that we get is actually nine

so if you go in your calculator and take

27 and raise it to the 2/3 power the

exact you know if you put the fraction

as in exact 2/3 and raise it like this

you're gonna get a 9 if you take 27 and

square it and then take the cube root of

that you're gonna get 9 there but I

mentioned that when we did the squaring

operation first it became cumbersome

because this number is big and then we

have to find the cube root of that

really big number so we can do it

another way we can do or the following

we can say that 27 to the 2/3 power

instead of squaring it first we can do

the other operation first we can do that

because of the way the exponents work we

can raise it to the 1/3 power and then

square the result how do we know we can

do that again because if I multiply

these exponents together I get exactly

what the problem statement was so now I

have to take the cube root of 27 so

let's just be explicit and write it down

27 cube root of this and then the result

of that I have to square it now the 27

is a whole lot easier to take the cube

root of right because you know that 9

times 3 is 27 and you know that 9 is 3

times 3 these are things I have in my

mind I don't know that 9 times 81 is

this I have to probably use a calculator

for that but this is actually easy and

I'm looking for triplets and I found a

triplet of threes and so what's gonna

happen is this in the middle is going to

become a 3 and then I'm going to be

squaring that and so I'm gonna get a 9

and that's the answer and notice that

this 9 is exactly the same is this one

because it doesn't matter the order in

which you do it that's what I was trying

to show you here in terms of variables

that you get the same thing no matter

what you do that's what this is trying

to tell you when you have anything

raised to a fractional exponent you can

either raise it to a power and then take

the root or you could take the root and

then raise it to the power same exact

thing that's what we did raise it to a

power then take a root that's what we

got take the root then raise it to the

power that's what we got same exact

answer all right

usually it's going to be easier on your

on yourself if you have the choice just

to go ahead and take the route first

notice that we took the the cube root

first and that was easier we knew the

cube root of 27 and then we could cute

the square and then we got the answer

going this way we're kind of required a

calculator or a lot of work on your

separate sheet of paper so if you have

the choice of which one to do first

usually you should go ahead and do the

radical first cube root square root 4

through whatever you have and then do

the other thing later all right so let's

keep on going let's say we have the

problem 16 to the 3/4 power again we can

either cube it first and then take the

root or we can take the root fort first

and then we can cube it but we just

learned that it's probably going to be

easier to write it like this 16 so the

1/4 power will do the radical first and

then we're gonna cube the result how do

I know this is legal because if I

multiply these exponents I get 3 times 1

and 1 times 4 I get 3/4 so this is

exactly equivalent to this and I know

that the fourth root I mean the one

fourth power is the fourth root of 16 of

course I still have to cube it and then

I know how do you take the fourth root

of 16 well you just go down here and say

well I know the 16 is 4 times 4 I know

that 4 is 2 times 2 and this is 2 times

2 and since it's a fourth root I'm not

looking for pairs or triplets I'm

looking for copies of 4 and so I found

that I have a 2 that I can pull out of

that radical but then that was just

under here I still need to cume it 2

times 2 is 4 and then the 4 times 2

again is 8 so 2 cubed is actually 8

that's the final answer now I'm choosing

to do the radical first because I have

choices if I were to do the problem

again of course I could take 16 and I

could cube it first but that's gonna

give you a big number and then the big

number is gonna be you're gonna have to

take the fourth root of that under a

radical do a big factor tree it's gonna

be a little more work so if you can it's

better to go ahead and do the radical

first it saves you a little bit of work

all right crankin right along we only

have a few more of these what if we have

negative 125 raised to the power of

negative 1/3 so notice that everything

in this parenthesis is raised to this

power and this power is itself negative

so that means everything in here is

under kind of like the spell of the rat

of the exponent here so because it's

negative we're gonna drop everything

below and we're gonna make it negative

125 to the positive 1/3 the negative

comes with it because it's in

parentheses and the 1/3 of negative 1/3

exponent applies to the whole thing so

you drop the whole thing down make it a

positive exponent all right and then we

know that 1/3 power means negative 125

is a cube root cube root goes there

because that's a 1/3 tower and you might

say well wait a minute I thought we

couldn't do radicals of negative numbers

well it is true that if you take the

square root of a negative number it's

the answer is not real it's an imaginary

number but this is not a square root

it's a cube root and the cube root of

negative numbers does exist and let's

see how you know that is the case so if

this were just a positive 125 the way

that you would write it is you would say

5 times 5 times 5 5 times 5 times 5

because this is 25 and then 25 times 5

is 125 but this isn't quite right

because you have a negative there but if

it's negative 5 times negative 5 times

negative 5 under this factor tree they

all multiplied together think think of

it this way negative times negative is

positive

but then positive times negative makes

it negative again so this times this

times this actually does equal negative

125 and that's why it does exist and so

what you're gonna get is 1 over negative

5 or if you want to be better about it

write it as negative 1/5 with the

negative sign because kind of sitting

out in the front there all right so

again when you have square roots of

negative numbers you don't get real

answers you get imaginary but cube roots

perfectly fine to take the cube root or

a fifth root or a seventh root any odd

power or any odd root you can take those

of negative numbers No

okay two more what if we have four to

the negative 0.5 as an exponent now a

lot of students will freeze up when they

see that because they see a decimal

there and they're like what do I do

well this is an exact number zero point

five is not an approximation it's

exactly equal to negative one half and

once you have it written like this you

can drop it downstairs to be positive

one half and then when you have it down

here you write it as the square root of

four because one half becomes a square

root and then the square root of four is

of course 2 so you get the answer of one

half so if you see a decimal that's

exact like that just change it into a

fraction and kind of works RIT now this

following one will be our last problem

what if we have negative 8 to the 2/3

power how do we simplify this guy well

there's a big gotcha in this problem and

you need to understand what it is you

see here notice whenever we had this

wrapped in parenthesis that we said the

negative 1/3 power applied to the whole

thing because including the negative

sign because it's wrapped inside

parentheses so we kind of had to bring

the negative along with it however there

is no parentheses here so a lot of

students will try to apply this exponent

to the negative that's here but that's

actually not right

because there's no parentheses there so

that negative is multiplied out in front

but that exponent does not apply to that

because there is no parentheses grouping

them all together so really a better way

to write this is you kind of put the

negative outside open the parentheses

bring the 8 here and make it 1/3 power

because we're gonna do the root first

and then square it like this so you see

the negative is not participating in the

exponent because it's just kind of

sitting as a coefficient in the front

the exponent is only applying to the 8th

so we kind of wrap it in parentheses to

kind of force myself to recognize that

and then this becomes a cube root so on

the inside I'm gonna have a cube root of

8 and I have to square the result now

what's the cube root of 8 you all know

that 8 is 2 times 4 and 4 is 2 times 2

and I'm looking for triplets because

it's a cube root so there's my triplet

and so what I'm gonna have

is the negative sign is still there

inside the parentheses I just have a two

because the cube root of three is two I

have to square what is inside those

parentheses the negative sign comes

along the ride but now I I square the

two and I get a four the negative sign

just stayed in front the whole time the

answer to this guy is actually negative

four if you go in a calculator or

computer and you put a negative eight to

the power of 2/3 you're gonna get a

negative four if you get anything other

than negative four then you put it into

the computer wrong because this negative

does not participate in the exponent

here if I had wrapped a parentheses

around the negative and around the eight

so that the whole thing was encapsulated

to the two-thirds then definitely it

would it would have been different

answer we're gonna have some problems

like that in a minute

but it would have been a different order

of operation so basically it's not

participating because it's not wrapped

like that so the most important thing

for you to learn in this lesson or to

understand is that fractional exponent

exponents because rational number is a

number that can be written as a fraction

basically the bottom denominator of that

fraction determines what route you're

going to be taking and the top of the

fraction determines what power the order

of that you do the root and the power

can be whatever order that you want

because of the way exponents work

because when you raise a power to a

power it doesn't matter the order in

which you do it

so usually though it's going to be

easier for you to take the root first

before raising the results of the power

as far as like how much work you have to

do so try to do that if you can follow

me on to the next lesson we have several

lessons here to get more practice with

rational exponents so make sure you can

solve all of these and follow me on to

the next lesson we're going to conquer

the rest right now