Up next in 10
Powers And Exponents
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Apr 1, 2025
Welcome to section 1 of chapter 9, in this lesson, we'll learn how to write expressions using exponents and also learn how to evaluate expressions containing exponents. Chapters: 00:00 Introduction 01:35 How To Write Numbers Using Exponents 04:13 how to evaluate negative numbers with exponents 08:34 The Importance of the parenthesis 10:45 Writing expressions using exponents 12:27 Evaluating expressions containing exponents
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0:00
so we were going to do a review of the
0:03
test but we're not going to do that till
0:05
next week so I postpone the test in next
0:07
week so that's what we're going to have
0:08
chapter eight test because of the other
0:10
class wasn't ready and I I believe going
0:13
to that thing yesterday kind of like
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stalled so we're going to talk about
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exponent okay so we're going to talk
0:20
about powers and exponent this is
0:23
chapter 9 what is it is it next week
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Iowa testing I don't know after I think
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it's next we next so we're going to talk
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about powers and exponent this is
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probably the one of the easiest uh
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section in chapter 9 right so for
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example an expression like 4 * 4 * 4
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that can be written right as what if you
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was to write 4 * 4 * 4 using exponent
0:53
and Powers it will
0:55
be three right but you have to make sure
1:00
the factors are equal right if they're
1:02
not equal you can't do that now some
1:05
people like make this mistake I'm saying
1:08
5 * 5 5 S is 10 so you have to be be
1:12
sure not to do that a lot of people when
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they see this 5 two they go oh that's 10
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that's not true right 5 squ is what 5 5
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* 5 that is 25 so do not fall into that
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category of those that can differentiate
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between five and five and 5+ 5 right now
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when you see an expression like this 4
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the 3r this is called the base and this
1:37
is called the exponent or the power
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right so this is called the base this
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called is the base and this is called
1:44
the power or the exponent okay now if I
1:48
was to say to you you can write an
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expression or write a number using part
1:53
of exponent how would you do that for
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example if I have
1:57
five five five and five and five and
2:00
five multiply what does that give you 5^
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5 right so pretty simple St this is 5 to
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the^ 5 right now it's easy when you have
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like positive integers now it gets a
2:13
little bit hard when we have like TR
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rational numbers or real numbers and uh
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negative integers here I have8 time8 * 8
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what do you notice about this they're
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all negatives they all negatives what
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else do you notice um three there are
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three of them what they are in
2:34
parentheses there's a reason for it
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right that's a reason for it theth is
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for a reason so if I was to write this
2:41
use an exponent what would I look
2:44
like it would be
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what so how would you
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write like
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this no you're you're acting like that's
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wrong probably is would you write like
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this parentheses you got to keep it in
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parentheses that's the reason why
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right so we're gonna keep this in
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parentheses like this now I'll show you
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why right now let's let's pause for a
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second what
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is8
3:14
8 give me one
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second probably actually let's use a
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smaller number to explain this right so
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let's say I have -2
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to the^ 3 and then -2 to the^ 3 am I
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going to get the same outcome no no yeah
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yes we will and we'll find out why we
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find out when this is important when
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this is not important right so what's -2
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to^ 3 what is that going to give me oh
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wait no you just say see I just talked
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about that how is
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that no no 2 that's right so what's -2
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CU this way would be what -2 *
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what
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time right I put in parentheses that
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mean the whole this whole number is
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being raised to the power of three so
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you have to do it three times right so
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-2 this is going to give
4:17
you8 right now this is going to be what
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this is what so what is the difference
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to do this will be negative
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right now do you see the difference
4:29
between the two oh it's because only the
4:30
first one okay keep talking hey for
4:33
America first of all I have so let's not
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do that so the difference between this
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and this is important if you don't pay
4:41
attention to this it's going to bite you
4:42
later on and you be like I don't
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understand I'll be like you wasn't
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paying attention now -2 parentheses Cub
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is -2 in parenthesis right -2 * -2 * -2
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the reason why you have to repeat the
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negative is because it's -2
5:00
as a whole as a whole that is being
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raised to the third power right but here
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if you look at it this is -2 ^ 3 the
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negative only applies one time oh right
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because the negative is not being
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tripled you see what I'm saying the
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negative this is like saying I'm doing 2
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to the 3 so this is -2 * 2 * 2 it's
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still going to give
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mega8 but you have to be careful
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especially when you come when you come
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when it comes to like even numbers now
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none of you guys is taking notes I don't
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know why but I'm assuming that you get
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it I am so let me give you an example
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right say I have this
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here -4
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s and then -4 squar right I have these
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two numbers that I want to I want to
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find what they are I got4 in parentheses
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square and then4 S
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so what do you think4 squ is going to
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give
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me what's that going to give
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me4
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Square I I want you to
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expand right this is4 * -4 because it's
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in
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so4 * -4 right and that gives you what
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positive 16 what about4 2 what is that
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going to give you * 44 * 4 which
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is see the difference between the two
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right the negative here does not apply
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until you're done multiplying the number
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inside right this is where you have to
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be careful this is why I'm always
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promoting using
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pares okay so now you also have the same
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the same thing with fractions right so I
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have now2 * 1/2 * 1/2 what's that going
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to give me * if I was to write that
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using an exponent is like a whole now I
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want to turn this into a exponent right
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a base on an exponent what's that going
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to give
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me I mean to the power three you see
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what I do 12 to the^ three right now the
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difference here is what I put the 1 12
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in parenthesis right you have to be
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careful again so is this the same let's
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wait let's let's learn some something
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else again if I have this here right
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there's at
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the and then if I have
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this do I have the same expression is it
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the same
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thing is it going to give you the same
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thing yeah are you sure
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cutive it doesn't matter is it going to
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give you the same thing what is one 1
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over 5 squ to the power two how would
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you if I to explain that what would that
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be * right so 1 over 5 squ like that
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right gives me what 1 over 5 time what
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but what about 1 s over 5 what does that
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give me want give you the same thing no
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it's not is it well what would the other
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number be what's the difference between
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the two here what is it it's the the
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second one the square is only for the
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one on top the the second one the square
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is only for the
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[Music]
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the right you see
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so here you do not have it's not the
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whole number that is being raised to the
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power of two only the one so here you're
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going to get what 1 * 1 over what just
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five you see the difference on there
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right you have to very careful this
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parenthesis changes the entire problem
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it changes the way the problem looks if
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I have parentheses covering the entire
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fraction that means the entire fraction
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that has to be multiplied by itself
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twice right here the only one that has
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to multiply by itself twice is the
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one does that make sense okay so we need
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to really pay attention to those
9:09
details we have to really pay attention
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to the details now
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I'm now um continuing right now I want
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you to write an expression using
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exponent so we have on top here I have x
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* x * x * x * y y * y * y * y could you
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do x^ 4 * y^ 4 right x 4 y to four right
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so you can use the Expression use an
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exponent you can do this now what about
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the third one you have a b c b a c
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c uh I don't know so how would you do it
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you put the two a together or no CU
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they're too far away you can rearrange
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it right the first thing you do here is
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you rearrange it like you said so
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rearrange so I'm going to put the two a
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together right so I know I cancel this
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out and then B is the same way right B
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and B so I cancel out and how many C's
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do I have left three three C's right so
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C * C * C and then what is that going to
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give me
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then no what that going to give me a
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what the power and then C to the power
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three
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right right that be c^ 3 now if I look
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here right I have x -
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y * x - y * x - Y and I want to express
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this using an exponent what's that going
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to give me be X be x - y power 3 three
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right x - y to the power of wait so for
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like the second one Like A and B you
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would do like you rearrange
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like since there's two a you put two and
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since there's like two B's you put two
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in it yeah because it's two B's right B
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* B is 3 S and then 3
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C's any random number I just like 7,000
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okay all right so now the next thing we
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going to learn how to do is wait could
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there hey Mr yes could there be a
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negative power excuse me could there be
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a negative power that we're not going to
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see negative power to the next
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chapter all right we want to talk about
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negative power in the next chapter right
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now we're just doing this one simple
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together Ang heyy can you guys like
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seriously all right so now we're going
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to learn how
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to evaluate expressions
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right how do I evaluate an expression
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evate all right so what I have is first
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before we do that we need to be
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introduced to couple of thing any number
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any number right any
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number with an exponent of
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zero is always going to be equal to
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one okay I got to write that down my
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nose is
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bleeding your nose is bleeding
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so any number ra to the power of 0 is
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always going to be equal to one right
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for example if I have 2 the 0 that's
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equal to
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what
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1 a th000 to the 0 is one
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right and then 5 to the 0 is one right
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now what about 12 to the Zero 12 to the
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Z one one what about
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-3 to the 0 now you be careful here 1 1
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right
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wa-3 to the0 be why is 1 no there a
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negative sign there's a negative sign
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right
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well so what about if I have four like
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this to the zero over five what that
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going to give
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me one 1 over 5 right because 4 to 0 is
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1 right so 4 over 5 would be 1 over 5
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all right good so we introduce to the
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zero power right anything greater than
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zero is one now you do have to be
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careful about the sign that behind it or
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whether denomination parentheses or not
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now the next thing I want I want you all
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to learn is this so let's assume that we
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have an expression that we want to
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evaluate right so let's say I want you
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to
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evaluate X to 2 + y 3
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if x is = to
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3 and Y is = to -2 right so let's say we
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have this and we want to evaluate it we
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want to find what it is so how would you
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do it you
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put no no no no I me I want to pay
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attention so how would you do this yeah
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you say you put you put the three where
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the x is and the Y
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he's right I don't want to hear those
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things in there all right so we want to
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evaluate it when X is 3 and Y is -2 so
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that means I'm going to take the three
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and I'm going to replace it by X and I'm
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going to take the -2 and I'm going to
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replace it by y but to do that you have
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to be extremely careful right why
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because symbol in the sign symbols in
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the sign so to do this we always going
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to use parenthesis when it com comes to
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evaluation of an expression so basically
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instead of me saying 3 squ it's going to
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be like this 3 in parentheses raised to
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the power of two right
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plus -2 parentheses rais to the power of
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three right so this is how we're going
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to do this because we don't want to run
15:21
into the problem of like messing up with
15:23
our negative and positive okay so now 3^
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2 is equal to what
15:35
which is n right so this is 3 * 3 and
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now plus you see here is going to be
15:41
what -2 * what -2
15:45
* -2 you see what I did here I did not I
15:49
did not just write the number without a
15:50
sign without the negative sign because I
15:52
don't want to run into any problems
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right so now 3 * 3 is =
15:57
to and then this is -2 that's plus that
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would
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be8
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right 9 +8 because -2 * -2 is 4 4 * -2
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is8 so this is 9 +8 which is technically
16:14
9 - 8 that is equal
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to 1 right that's equal to one so this
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is how you evalu an expression so let's
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do a couple more problems like this and
16:25
then I'll see how you guys uh Fair
16:31
I
16:32
have
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than you see
16:44
how did you get
16:50
one I think can I please erer thank
16:57
you all right
17:01
so so let's assume that we are trying to
17:09
find so let's say that we trying to find
17:12
this here right
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wait you in right now for uh seventh
17:16
period Spanish do you know what that
17:19
means I can switch to Spanish yeah you
17:22
here God I will I'll have so much
17:26
fun'll miss you but I don't really think
17:28
that
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yeah my art
17:34
are what do you
17:39
guys I need a
17:44
marker okay that's why you used it is it
17:47
still recording
17:52
yes all right so we're trying to find
17:54
this expression okay we're trying to
17:57
evaluate the expression using what we do
18:00
want all right so I
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have I have three Expressions that I'm
18:05
trying to evaluate and I'm giving their
18:08
values right first one is x * y Cub y 3
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right X is -2 and then Y is pos3 so how
18:23
would you write that no I don't that's
18:24
not what I'm asking I'm I already have I
18:27
have the value right so how would you do
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this x is -2 Y is 3 I want to find x * y
18:33
3 power oh I already got
18:44
this there you go there's no parentheses
18:47
around all right there's no parentheses
18:49
but what did I teach what did I say we
18:51
don't when you have X and Y so first you
18:53
have to write it correctly right we
18:56
start by writing the stuff correctly so
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X is equal to what 22 so I put in
19:02
parentheses right2 and then Y is = to
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three now you see what I did here which
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one is being raised to the^ three is it
19:11
X Y or just
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x y y so you see what I did is -2 in
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parenthesis *
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3 to the 3 now -2 is just2 right we just
19:23
need to work on the next one what is 3
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to the thir is 3 27 * 3 * 3 and this is
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now -2 * what 27 27 which
19:36
is
19:39
Nega 54 54 right so do you see how we're
19:43
doing it we have to really be careful
19:46
with the up uh the order of operations
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this is the whole reason why we doing
19:52
this section understand this I'm
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goodness my favorite Tik tocker she's in
19:56
labor and she's about can you not
19:59
mention that because I this is
20:01
publicized I don't really care
20:05
[Music]
20:08
all that out all right so now we have x
20:12
x 2 + 2 y - 3 right so what do I know
20:17
exced so I have X to
20:20
2 + 2 y minus three she I don't care
20:26
she's doing on purpose so what do I
20:28
replace I push X by
20:30
what and I put in parth good job on that
20:34
I like how you put it in parentheses
20:36
good and then plus 2 Y 2 * what three
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3
20:42
minus three right now what is -2 to the
20:46
power two what's that going to give you
20:48
be -2 * -2 would be 8 no no it be four
20:53
four is it negative or positive positive
20:56
positive right this is 2 *
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-2 + 2 * 3
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right there you go so this is four here
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3 so 7 + 6 and then minus 3 that's right
21:10
that will be seven right 4 + 6 is 10 10
21:14
- 3 is 7 okay good now the last one here
21:19
I have 7 * X2 so what is that equal how
21:23
would you set that up uh so if the so it
21:26
would be seven m parentheses
21:30
-2 other parentheses the power two so it
21:36
would be four no it be 7 * 4 7 * 4 which
21:40
is 28 28 good all right good that's
21:45
good all right so now now I'm going to
21:48
challenge I'm going to challenge you a
21:49
little bit right uh so now let's say I
21:52
have this here
21:59
I smell sour for some
22:06
[Music]
22:11
reason so
22:13
rude grou the all right so how would you
22:16
do this we have x to the Z right plus 2
22:21
y to the
22:24
Z is 1 2 * 1 + what 2 * 1 * 1 - what - 5
22:32
so here we didn't even need to like
22:34
replace the X and Y by their values
22:36
because we already know what they are
22:37
right yeah so therefore that gives you
22:39
what -2 that's correct all
22:43
right and now let's let's try this one
22:49
here so I want you to do this
22:52
one I want I want you to find this
22:55
number for me
22:59
don't be
23:02
mean so actually let's start with each
23:05
some simple so -2 5 squ so I want you to
23:08
actually evaluate the number tell me
23:10
what it is in terms of what is the
23:12
number right keep it as a fraction but
23:15
simplify fraction if
23:23
possible can you change the exponent did
23:27
mm oh
23:33
don't you
23:49
cross all right
23:51
[Music]
23:53
good what's five five I told you all
23:57
that right that too so so this is -2 *
24:00
-2 over what 5 * 5 which is what 4 over
24:04
25 now we going to do this one and I'll
24:07
set I give you guys work okay I 4 x + 1^
24:11
4 what's that equal when X is -3 four *
24:16
and then
24:20
four3
24:22
first I haven't it
24:29
okay wait oh you do that
24:34
too
24:49
okay all right
24:52
good so we're going to do what's in the
24:54
pares first because of pendas right -2 +
24:57
1 is 2 to the^ 4 and now we have four
25:01
and this is what
25:02
-2 -2 -2 -2 so this here this is four
25:11
right is another four ex so it's
25:13
basically 4 * 16 right which is 64 right
25:18
yeah all right 64 okay now let's let's
25:21
get you some work and then y'all can we
25:24
can call you this so