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>> Shannon: Good morning.
This is Professor Shannon
Gratsy [assumed spelling]
for Americas to College.
We are working
on Robert Blitzer's
Introductory
and Intermediate Algebra
textbook, Section 5.7.
This will be the last section
in Chapter 5,
and we'll be moving
into factoring next.
We'll be covering negative
exponents and scientific
notation in this lesson,
and we will cover using the
negative exponent rule,
simplifying
exponential expressions.
Now you're going
to have all the rules
so you'll be able
to completely simplify.
We'll be learning
about scientific notation,
converting from scientific
to decimal notation,
and then from decimal
to scientific, and then work
on computations
with scientific notation
and some applied problems.
So here we go.
Why don't you guys warm up?
You can do the warm
up whichever method
you prefer.
Notice that the divisor is one
that you can use synthetic.
Ok, let's see how you did
on the warm up.
We have...
I'm going to put the x plus 3
in the house,
and then we've got our 7x
to the fourth,
zero x cubed-be careful-zero x
squared and then minus 8x
plus 0.
Now to match that 7x fourth,
we need to multiply x
by 7x cubed.
So this is going
to yield a result of 7x
to the fourth plus 21x cubed
and then subtracting
that binomial,
we will get negative
21x cubed.
Bringing down the 0x squared,
we will have
for our next division,
we need to match the negative
21x cubed.
So we need a minus
21x squared.
This gives us our minus 21x
cubed and then minus
63x squared.
Subtracting
that binomial turns them
both positive.
So we'll get our 63x squared.
We're bringing
down a minus 8x,
and to match the 63x squared,
we add 63x.
Multiplying through,
we'll get 63x squared plus...
So 3 times 3 is 9,
and then 3 times 6 is 18.
So we'll get 189x,
and that will be positive,
and then subtracting the whole
thing, we will have-the signs
will end up being the same
here-so we'll end
up with negative 197x,
and then bringing down the 0.
To match that we just need
minus 197.
We get negative 197x,
and when I multiply the
197 times...
negative 197 times the 3,
I will get minus 21,
and then 3 times 7 is 27...
29... so negative 591.
So when we subtract,
our remainder will be a
positive 591.
So at the end of the day,
our result
for this problem equals...
let's see what we got here...
7x cubed minus 21x squared
plus 63x minus 197
and then plus 591
over x plus 3, and we're done.
How'd you guys do on that one?
Good. All right.
Next step,
we need to simplify.
Ok? So here what'll happen is
we get 1 over,
and then using the product
powers rules,
we'll get 6 squared times x
to the 3 times 2,
which is 1 over 36x to the 6.
Ok, so moving right along,
we start going
into negative integers
as exponents.
A non-zero base can be raised
to a negative power.
The quotient rule can be used
to help determine what a
negative integer
as an exponent should mean.
So, I mean, one way to look
at this is
if you have something like,
I don't know, 2 cubed over 2
to the fourth, and if we just,
you know, use the definition
of exponentiation-the
numerator's 2 multiply
to itself 3 times.
Denominator is 2 multiplied
to itself 4 times.
So if you notice, you know,
each of these 2's divide
out to be 1, and we're left
with 1 over 2 to the 1, which,
if we took a look
at what was going on,
we could also think of it
as 2 cubed over 2
to the fourth is equal to 2
to the 4 minus 6 using the
quotient rule.
Oh, no, I'm making things
up there.
Two to the 3 minus 4 using the
quotient rule,
which would give us 2
to the negative 1.
Now looking at this further,
right, isn't 2
to the negative 1 the same
as 2 to the 0 minus 1?
And if we use
that quotient rule backwards,
that would give us 2 to the 0
over 2 to the 1,
which is one-half.
Ok, so however you want
to think about it, you know,
these are the...
these are negative integers,
right, and we'll be working
with them.
So here we go.
If b is any real number other
than 0 and n is a natural
number, then b
to the minus n is equal to 1
over b to the positive n,
and then next step,
negative exponents
in numerators
and denominators.
So if b is any real number
other than 0
and n is a natural number,
then just as above,
b to the minus n is equal to 1
over b to the n. Now
if you have a negative
exponent that's
in the denominator,
like 1 over b
to the negative n,
then that would end
up having a result of b
to the n over 1,
which is just b to the n.
When a negative number appears
as an exponent,
switch the position
of the base from denominator
to numerator or from numerator
to denominator...
[ Pause ]
and make the
exponent positive.
So remember nothing
in math is free.
So if you're just swapping,
you know, the base
from denominator to numerator
or numerator to denominator,
you pay for it
by making the
exponent positive.
The sign of the base
doesn't change.
So just because you have a
negative exponent doesn't mean
you have a negative number.
Ok? All right,
so here's some examples.
Again, we have to follow all
of our orders of operation,
order of operations,
and just take it step by step.
So here we go.
The first one, now,
the negative is not raised
to that power.
So this is equivalent
of the opposite of 1
over 7 squared.
So the change that I made,
7 to the negative 2,
is equal to this,
and then that gives me
negative 1 over 49,
and I'm done.
So the reason why it was
negative was
because this negative was
being multiplied
to the coefficient.
Now let's compare
to Part B. Notice the negative
is now in the parentheses.
So this is equivalent to 1
over quantity negative
7 squared.
So for this one,
this whole thing is equivalent
to this, and that
in turn gives us a result
of positive 1 over 49
because the negative is
being squared.
So it goes away.
All right?
Ok. So these...
this one here 3
to the negative 1 is 1 over 3
to the 1 minus 6
to the negative 1 is 6 1
over 6 to the 1.
So this guy was this,
and this guy was this.
So that, of course,
is just one-third minus
one-sixth, and what do we do
to simplify that?
Good. We need a LCD.
So 6 is the LCD,
and I multiply that one-third
by 2 over 2.
So we'll get two-sixths minus
one-sixth, which, of course,
is one-sixth, and we're done.
Ok, so Part D. Part D will be
1 over x to the twelfth
divided by 1 over y to the 1.
So here was the x's,
and here were the y's.
Put my equals
in the middle there.
So this will give us 1 over x
to the twelfth
and then times y over 1.
I'm inverting and multiplying,
which is y over x
to the twelfth.
All right.
So here we go.
Simplifying
exponential expressions.
Properties
of exponents are used
to simplify
exponential expressions.
An exponential expressions is
simplified when-so be careful
that you understand this
because when I say simplify
on the test,
this is what I mean.
Each base occurs only once.
No parentheses appear.
No powers are raised
to powers.
No negative
or 0 exponents appear.
[ Pause ]
Ok. Steps for simplifying
exponential expressions.
If necessary,
be sure that each base appears
only once using b
to the m times b
to the n equals b
to the m plus n, or b to the m
over b to the n equals b
to the-good-m minus n.
If necessary remove
parentheses using the product
to powers rule
or the quotients
to powers rules.
[ Pause ]
If necessary, simplify powers
to powers using b
to the m raised
to the n equals b
to the m times n.
If necessary rewrite
exponential expressions
with 0 powers as 1.
Remember that some base
to the 0 is 1.
Furthermore, write the answer
with positive exponents using
b to the negative n equals 1
over b to the n, or 1 over b
to the negative n was equal
to b to the n. All right.
So here we go.
Why don't you go ahead
and pause the movie
and see how you do
on these problems.
On your mark, get set, go.
[ Pause ]
Let's check your work.
So here we go.
This is...
Remember how we do 45 divided
by 15, and then we're going
to have z to the 4 minus 12
using the quotient rule.
Forty-five divided by 15 is 3,
and we will get z
to the negative 8.
So here's that extra layer.
This is equivalent
to 3 times 1 over z
to the eighth.
So this here is this,
and then we can finish it off
by writing it
as one fraction-3 over z
to the eighth.
Ok. Next step, we've got,
you know, a product
to powers rule.
So this is going
to be 3 cubed,
and then raising a power
to a power, we'll have y
to the 4 times 3,
and then we have...
I'm going to do this y
to the negative 7 minus 7.
So here we go.
This here is this portion
of it, and then this here was
this portion of it.
So this will give us 3 cubed
is 27.
We're going to have y
to the twelfth,
and then times y
to the negative 14.
So now working on the y's,
we've got two
of the same base.
So y to the 12 plus a negative
14 using the product rule.
So this portion here is shown
here, and then we will get 27y
to the negative 2,
which is 27 times 1
over y squared,
which gives us 27
over y squared.
So, again, we've got our y
to the negative 2.
Let me find a different
color here.
So this y to the negative 2 is
the equivalent to 1
over y squared.
All right.
So, again products to powers.
So 5 squared and then x
to the 3 times 2
over x to the 7.
So we will get 25x to the 6
over x to the seventh,
which will give us 25
and then times x
to the 6 minus 7.
We'll get 25x
to the negative 1,
which is 25 times 1 over x
to the 1, which is 25 over x.
[ Pause ]
And let's just show you how
it, you know, matches up.
So x to the sixth over x
to the seventh gave us x
to the 6 minus 7,
and then which of course,
gave us x to the negative 1,
which gave us this.
Ok. Last step.
This is quotients to powers.
So this one's going to be x
to the 3 times negative 4
over y to the 2 times negative
4, which will give us x
to the negative 12 over y
to the negative 8,
which gives us 1 over x
to the twelfth over 1 over y
to the eighth, and that,
in turn, when we invert
and multiply gives us y
to the eighth over x
to the twelfth.
[ Pause ]
All right.
So trucking right along,
we use negative exponents
in scientific notation.
So here we go.
A positive number is written
in scientific notation
when it is expressed
in the form a times 10
to the n where a is a number
greater than or equal to 1
and less than 10.
So in mathese, 1 is less than
or equal to a, which is less
than 10, and n is an integer,
and we use scientific notation
to make either very,
very big numbers
or very small numbers easier
to write out.
It's kind of an abbreviation
for numbers.
So it is customary
to use the multiplication
symbol, x, rather than a dot
when writing a number
in scientific notation.
We can use n, the exponent
on the 10 in, you know,
the general form,
a times 10 to the n,
to change a number
in scientific notation
to decimal notation
if n is positive.
Move the decimal point in a
to the right n places,
and if n is negative,
move the decimal place...
decimal point in a to the left
by n places.
So here we go.
Write each number
in decimal notation.
Well 7.85 times 10
to the eighth.
We are increasing 7.85
by 8 places.
So this will be 7...
Ok, now 8, I've moved it once;
5 I've moved it twice;
and I'm going to need one,
two, three, four, five,
six zeroes.
So it turns
out that 7.85 times 10
to the 8 is 785,000,000.
Ok? Now this one,
notice the decimal is implied
behind that 9.
So this is equal to...
I have to move it 5 places
to the left.
So I'll just leave some space.
So if it's here,
it'd be one place
and then two places,
three places, four places,
five places.
Now I'm going
to put this 0 here
so that you can more easily
see where the decimal is.
All right,
you guys try Parts C
and D. See how you do.
[ Pause ]
All righty.
Let's see how you did.
This one, 2 is positive.
That power
on the 10 is positive.
So I if I take my 1001,
I need to move my decimal two
places to the right.
So I get 100.1.
This guy, I need
to move the decimal one place
to the left.
So we will have
9,999 ten-thousandths.
All right.
Moving right along.
Converting from decimal
to scientific notation.
First off, write the number
in the form a times 10
to the n. Remember a is only
going to have, you know,
it's going to be in the 1's.
All right?
Determine a,
the numerical factor.
Move the decimal point
in the given number
to obtain a number greater
than or equal to 1--
this is working
on our a-and less than 10,
but nothing in math is free.
So you have to figure out,
you know, how you changed it.
All right.
So this is
where you determine n,
the exponent on 10 to the nth.
The absolute value
of n is the number
of places the decimal point
was moved.
The exponent n is positive.
If the given number is greater
than 10, and negative,
if the given number is
between 0 and 1.
Ok? Ok. So here we go.
Part A, we're determining the
a factor.
So we have 6589.
I need this to be a number
in the 1's.
Where does my decimal go?
Good. It's got to go here.,
all right.
6.589... now times 10.
Now I have to figure
out the n.
So the decimal place was
moved, right,
to get from the given number
to the 6.589.
I moved this decimal place
one, two, three, four, five,
six, seven,
eight places, right?
One, two, three, four, five,
six, seven, eight places.
All right,
so there was a movement
of 8 places.
Now this number was
between 0 and 1.
It's very small, right?
So the 8 has
to have a negative on it.
And we're done.
Ok, so you guys give B, C,
and D a try.
Pause the movie and give B, C,
and D a try.
[ Pause ]
All right,
let's see how you did.
So this one, the numbers 6, 7,
8, and 9 are the ones
that matter.
Ok, and then I need the
decimal between the 6
and the 7 to make it a number
that's between 1 and 10,
not including 10.
Now I need to figure
out the exponent.
Now this number's a very,
very large number.
The decimal was here, right?
So 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12 places was how many
places it moved,
and the 12 is going
to be positive
because this is a very
large number.
So in order
to get an equivalent number,
you'd have to multiply by 10
to the twelfth to get it back
to what it was.
All right,
so Part C we have the numbers
2, 3, and 4.
Where does the decimal go?
Good. In between that 2 and 3,
all right,
and then this is a number
between 0 and 1.
The decimal place was
moved once.
So I will have a negative 1
as my decimal.
All right,
and then last problem here.
We've got to keep
that 2, 3, 4, ok.
So these 0's need to stay
in there for the scientific
notation, but the 0's behind
it we can take care
of by our times 10
to the power.
So our decimal needs
to go here,
and then times 10 to...
the decimal number
started here...
1, 2, 3, 4, 5, 6, 7, 8,
9 is the number
that is larger than 1.
You know, it's
like I have a
positive exponent.
[ Pause ]
Ok. So computations
with numbers
in scientific notation.
All right.
I mean they follow all the
rules for exponents.
So if you have, you know,
a times 10
to the n times-remember these
x's are not the variable x
for multiplication
because we're dealing
with scientific notation-b
times 10 to the m. We're going
to have the a times the b
and then times 10
to the n plus m
because it's all
being multiplied.
All right, so for division
if we have a times 10
to the n divided by b times 10
to the m, we will get a
divided by b times 10
to the n minus
m. Exponentiation.
If you have a number inside a
scientific notation
and it's being raised
to some power, this is going
to end up being a
to the m times 10
to the n times m. Now you have
to be careful, all right?
After the computation is
completed,
the answer may require an
additional adjustment before
it is expressed
in scientific notation.
So you do all the work,
and then you take a last look
at it and make sure it's
in scientific notation.
Ok, so Part A. This is equal
to 3 times 4 times 10
to the 4 plus 2.
This gives us 12 times 10
to the 6.
Am I done?
No. Right?
I need to work
on this 12, right?
So I'm going
to have 12 times 10
to the something,
and that times 10
to the 6 is going
to be hanging out.
So I'm basically working
on this one.
So I need to do 1.2
to get this
into scientific notation.
I moved this one place, right?
One place and 12 is greater
than 1.
So 1.2 times 10 to the 1,
that's equivalent to my 12,
and then now I'll have 1.2
times 10 to the 1 plus
6-combining 10
to the 1 times 10 to the 6,
and I'll get 1.2 times 10
to the seventh.
[ Pause ]
Ok. This next guy,
raising it to the power,
so we'll have 2
to the fifth times 10
to the negative 3 times 5.
This gives us 32 times 10
to the negative 15.
Am I done?
No. I've got to work
on that 32.
32 is the same as 3.2 times 10
to the...good...
1, and I still have the times
10 to the negative 15
hanging out.
So 32 is equivalent to this.
All right,
and then I will get 3.2 times
10 to the 1 plus a negative
15, which is 3.2 times 10
to the negative 14.
And we're done.
Ok. So two more problems.
Why don't you guys pause the
movie and give these a try?
All right.
[ Pause ]
All right.
So let's see how you did.
This one here,
we've got our 180 divided
by 2 times 10
to the-good-8 minus 4.
This will give us 90 times 10
to the fourth,
but the 90 I've got to switch
to 9.0 times 10 to the 1.
Then I've got the times 10
to the fourth hanging out.
So 90 is equivalent to this,
which, of course,
just gives us a 9 times 10
to the 1 plus 4,
which is 9 times 10
to the fifth.
And Part D, this gives us 5
to the negative 1 times 10
raised to the 4 times
negative 1.
Five to the negative 1 is
one-fifth times 10
to the negative 4.
One-fifth,
we need to write this
as a decimal.
So one-fifth is 0.2,
and then we have our times 10
to the negative 4.
Are we done?
No. We've got
to change this two-tenths
and make it a 2 times 10
to the negative 1 times 10
to the negative 4.
So 0.2 is equivalent
to this yellow here.
This will give us 2 times 10
to the negative 1 plus
negative 4,
and we'll get our result
as 2 times 10
to the negative 5.
We're all done.
Ok. So some applications.
A human brain contains 3 times
10 to the tenth neurons,
and a gorilla brain contains
7.5 times 10
to the ninth neurons.
How many times
as many neurons are
in the brain of a human
as in the brain of a gorilla?
So we're doing ratio of human
to gorilla.
So human to gorilla, ok?
So here we go.
The human brain had 3 times 10
to the tenth neurons, right?
So that's our human stuff,
and the gorilla brain had the
7.5 times 10 to the ninth.
So then this is equal to,
if I do...
So I need to do 3 divided
by 7.5 and then times 10
to the 10 minus 9.
Well, 3 divided by 7.5 turns
out to be 0.4 times 10
to the negative 1.
Converting
to scientific notation
because .4 is smaller than 1.
So I will get 4 times 10
to the negative 1 times 10
to the...
Oh, and here I apologize.
I hope you caught that.
This here was a positive 1.
So 10 to the positive 1,
and this here is equivalent
to this.
So now we'll get 4 times 10
to the negative 1 plus 1,
which is 4 times 10 to the 0.
So at then end of the day
on this one, we're just going
to get 4 because 10
to the 0 remember is 1.
All right.
The second application.
If the sun is approximately
9.14 times 10
to the seventh miles
from the earth,
how many seconds
to the nearest tenth
of a second does it take
sunlight to reach the earth?
Use the motion formula,
distance equals rate times
time, and the fact
that light travels at the rate
of 1.86 times 10
to the fifth miles per second.
So let's see.
We have our distance is 9.14
times 10 to the seventh,
and then we don't know the
time, but we do have a rate
of 1.86 times 10 to the fifth.
So distance equals rate
times time.
Well we have the distance
and the rate, so do you see
if I isolate t I will get
distance divided by rate?
So continuing,
I'll have my 9.14 times 10
to the seventh
and then divided
by 1.86 times 10 to the fifth.
So the time will be 9.14
divided by 1.86 times 10
to the 7 minus 5.
We'll have our time of 9,
and they wanted us to round
to the nearest tenth
of a second.
So we'll end
up with 4.9 times 10 squared,
and then answering the
question we will say,
"It takes
sunlight approximately...
[ Pause ]
4.9 times 10 squared seconds
to reach earth."
[ Pause ]
And actually, you know,
up here we should have
answered the question,
too, right?
So we should've said here...
We should've said,
"Humans have 4 times
as many neurons ...
[ Pause ]
as gorillas in the brain."
All right?
And that wraps it
up for today.
I hope you guys have a
fabulous day.
Don't forget to study hard.
Do your homework
and take the Chapter 5 quiz
as soon as possible
after you're done
with the section.
All right.
Bye.