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## Exponents with negative bases

Current time:0:00Total duration:8:39

## Video transcript

- [Voiceover] Let's now
do some examples that test our understanding of what happens when we multiply or divide a bunch of positive or negative numbers together. And like always, pause the
video and see if you can work through it on your
own before I do it. So first question: What is the
sign of s to the 67th power divided by t to the
ninth power, divided by the expression 3s times s over 4t, when s is less than zero,
and t is greater than zero? So the really interesting
thing here is that s is less than zero, s is negative. So if s is a negative number, we first have s to the 67th power. And we just have to remind ourselves, a negative number being multiplied, if you have an odd number
of negative numbers being multiplied together, you're going to have a negative number. And s to the 67th power, that's 67 ses, so it's literally s times s times s, and you literally are going
to have 67 of these ses, and you're multiplying them all together. So you have an odd number of ses, an odd number of negative numbers that you're multiplying together. So if you have an odd number of negative numbers multiplied together? Well that means that the
product is going to be negative. So this thing right over
here is going to be negative. Now what's t to the ninth? Well, t is a positive
number, they tell us that. They tell us that t is greater than zero. Well, a positive number,
really to any power, however many you have
that multiply together with an even or odd, if
it's just all positive, well this is going to be positive. So this thing right over
here is going to be positive. Now what's a negative
divided by a positive? Well a negative divided by a positive is going to be negative. We've already seen that show before. So this whole thing,
I'll circle it like that, is going to be a negative. And so that this thing
doesn't get too messy, actually let me just rewrite this. So all of this stuff that
we've written over here, I'll just write this
as this is going to be some negative number, and then we're going to divide it by 3s times s over 4t. Now this expression right over here. What is s divided by 4t going to be? Is that going to be positive or negative? Well we know that s is negative. T is positive, so four times
t is going to be positive. So this thing right over
here is going to be positive. Negative divided by positive, well that's going to be a negative. So actually let me just write it this way. So all of this stuff, I
could just replace with s divided by 4t is going to be a negative. So times a negative, times
some other negative number. Now what's 3s? Well s is negative, so 3s is negative. So that is negative, so you have a negative times a negative. Which is a positive. So all of this business right over here is going to be positive. Another way you could think about it is you're going to have a
negative times a negative, which is positive, and then
divide that by a positive, it's just going to be positive. So all of that, so this whole thing, all of
this business right over here, is going to be a positive. This is all a negative,
this is a positive. You're gonna have a negative
divided by a positive. Well that's just going to be negative. So the sign of this crazy
expression right over here? It's going to be negative
if s is less than zero and t is greater than zero. Let's keep going, this is fun. All right. So what is the sign of p times q over p, times four and 2/7, when
p is greater than zero, and q is equal to zero. And the way I just
stressed q is equal to zero should be a big hint. And let me give you another hint. Pause the video, if you really think about what I just said, that q is equal to zero, you should be able to do
this in less than a second. So why did I say that you could do this in less than a second? Well if you just have a bunch of numbers, and then you multiply them times
zero at some point in that, the whole thing is just
going to be equal to zero. I could literally have a
times b times c times d, I could multiply a bunch of numbers. And if I just knew that one
of these numbers is zero, then the whole product is
going to be equal to zero. You know if c is equal
to zero, I could multiply a times b times d times
e and get some number, and then multiply that times c, it's gonna be anything times zero is zero. Now notice, q is equal to
zero, and q is right over here. So I could take p divided
by p, that's gonna be one. One times q, that's just gonna be zero, as one times zero is zero, then you're gonna have
zero times four and 2/7, it's just all gonna be zero. And the key is, there's
a zero right over here. At some point you're multiplying this entire product times zero, so the whole product is going, or you're multiplying
these numbers times zero, so the whole product is going to be zero. So this side isn't positive
or negative, it's zero. All right, there's no sign there. It's neither positive nor negative. All right, let's do another one. What is the sign of negative 3/4 times negative a to the fourth over three, when a is less than zero? So they're telling us that a is negative. So a is negative. So let me use a more dramatic color. So right over here, I have a negative number to the fourth power. So it's a times a times a times a. And we've already said if
you have an even number of negatives being
multiplied by each other, then that's going to be a positive. You could even see it over here, I mean they're all
negative, but a negative times a negative is
going to be a positive. A negative times the
negative, I take the third and the fourth one here,
that's gonna be a positive. And then a positive times a positive, the whole thing is going to be positive, even though each of these are negative. Negative times negative, or
another way you could say it, a times a is going to be positive, then times a is going to be negative, but then you multiply times a again, it's going to be positive again. So a to the fourth power
is going to be positive. But then you have a
negative in front of it. So you're taking the negative of a positive divided by a positive, Well, so this, all of this stuff right over here is all gonna be positive, but then you have this negative in front. So this entire thing
inside the parentheses is going to be a negative. And then you have a
negative, negative 3/4, times a negative, well a
negative times a negative, the whole thing is going to be positive. So what's the sign of this? It's going to be positive. Let's do another one,
I'm having too much fun. All right, what is the sign
of x to the 59th power, divided by 2.3x times 4/5, when x is less than zero? So x is once again a negative number. So x to the 59th power,
that's an odd number of negatives being multiplied together. So this thing right over
here is going to be negative. That's going to be
negative right over there. And this x right over here
is going to be negative. And so 2.3 times that x,
well that's going to be negative as well, so this
thing is going to be negative. That's going to be negative. A negative divided by a
negative, is a positive. So all of this business
is going to be a positive. A positive times a positive, well that's just gonna be positive. So all of this whole thing
is going to be positive. I can't stop, this is, all right. What is the sign of negative
x, times y divided by 7/8? And here they're showing us
x and y on a number line. And the key takeaways here
is that x is positive, x is to the right of zero, y is negative. So let's think about this. X is positive, but then we're taking the negative of x right over here. So x is positive but negative
of x is going to be negative. You bring a negative number
in front of a positive number, that's going to be a negative number. Now what's y divided by 7/8? Y is a negative number. I'm dividing it by a positive number. So a negative divided by a positive is going to be a negative. So this whole expression,
this whole expression right over here, is going to be negative. And then we have a
negative, all of this stuff, times a negative, well
that's going to give us a positive. So the whole thing here
is going to be positive.