01 - Simplify Rational Exponents (Fractional Exponents, Powers & Radicals) - Part 1
Summary
TLDRThis 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
- ๐ The lesson focuses on rational exponents, which are exponents containing fractions.
- ๐ข Basic exponent rules are reviewed, such as squaring a number (5^2 = 25) and negative exponents (5^-2 = 1/(5^2)).
- ๐ Negative exponents are handled by inverting the base and making the exponent positive.
- ๐ฏ Any number to the zero power is defined as 1, a fundamental rule in exponents.
- ๐ The lesson transitions into more complex problems to prepare for studying exponential functions and logarithms.
- ๐ Rational exponents are introduced as fractions, where the numerator indicates the power and the denominator the root.
- ๐ 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.
- ๐ 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.
- ๐งฎ Practical examples are solved to illustrate the process of calculating expressions with rational exponents, such as 81^(1/2) which equals 9.
- โ 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
๐ 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.
๐ 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.
๐ข 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.
๐ 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.
๐ 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.
๐ 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.
๐ 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.
Mindmap
Keywords
๐กExponential Function
๐กLogarithm
๐กRational Exponents
๐กNegative Exponents
๐กZero Exponent
๐กSquare Root
๐กCube Root
๐กMultiplication of Exponents
๐กOrder of Operations
๐กDecimal Exponents
๐กImaginary Numbers
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
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