Welcome to a presentation on the properties of exponents. The goal of this video is to
use the properties of exponents to simplify expressions. Here is a list
of the properties of exponents we will be discussing today. You may want to pause
the video now and write these down. We will consider them one at a time.
The first property is a product property. It states that if you're multiplying and
the bases are the same, you add the exponents. Let's take a look at why that makes sense. If
we want to multiply five to the second power times five to the fourth, we know
that five to the second means five times five, and we know that five to the fourth
would be four factors of five. And so now we can just count the total factors of five,
and we can see we're going to have six factors of five.
Well, if we take advantage of the product rule, that states that we can just take the exponents
two and four and add them to get five to the sixth. Now this is obviously much shorter,
but if you're not sure of a rule, it's always best to expand the exponents to find the product. Here
we have x to the sixth times x. Well, the first thing we have to recognize is that when we have
x and there's no exponent listed, that means the exponent will be one, and that's actually one of
the properties listed on the previous screen. So if we have x to the sixth times x to the first,
if we use a product rule, we could just take the exponents and add them to get x to the seventh.
Hopefully, that's logical because we know if we expand x to the sixth, we'd have six factors of
x times an additional factor of x, which of course would give us a total of seven factors of x. The
quotient rule states that if we're dividing and the bases are the same, we subtract the exponents.
So for this first problem, if we have three to the sixth divided by three to the fourth, let's
assume that we could not remember the quotient rule. Again, we would expand each of these,
so we'd have six factors of three in the numerator and four factors of three in the denominator.
And now we can simplify. Every three over three would simplify to one, and now we can
see we would have three to the second power or three squared. Well, using the quotient rule,
what it's stating is we could have just recognized that the bases were the same and simplified it to
three to the sixth minus four, which of course gives us three to the second. But if we don't
understand why this rule works, it's best to expand it and simplify it the long way.
So for this next problem, you can see as the exponents get larger, it kind of gets
more tedious to write out fifteen factors of y and eight factors of y. So we will just apply
the quotient rule by taking the exponent in the numerator and subtracting the exponent
from the denominator to obtain our simplified expression of y to the seventh. In this last one,
we know that whenever we have something over itself, it is going to equal one.
If we applied the quotient rule to this problem, we would have w to the fourth minus four power or
w to the zero power, which is equal to one. And this is one of the properties that we saw on the
first screen. Anything raised to the zero power is equal to one. The power rule states that we have a
raised to the power of m all raised to the power of n. This will equal a to the m times n power.
Again, before we apply this, let's take a look at it by expanding this. If we have
three cubed raised to the second power, as long as we understand what exponents mean,
we can expand all of this and we should get the same answer that we would get from
the power rule. So if we know if something is squared, that's multiplied by itself two times,
and now we also know that two cubed means two times itself three times.
So there's two cubed times another two cubed, and now we can just count the factors and we have
two to the sixth. Now, if we try to apply the power rule, we have a power raised to a power,
multiply. Of course, two to the power of three times two, of course gives us
the same answer of two to the sixth. So if you ever panic and you forget some of these rules,
just expand your exponents, and you can still derive them the long way.
Looking at our second example, we have m to the fifth power raised to the fourth power. Well,
again, the shortcut would be to raise m to the five times four power or multiply our exponents,
which would give us m to the twentieth power. Next,
we have the product to power rule and also the quotient to power rule. So if we have
a times b raised to the power of n, that will equal a to the n times b to the n.
Now to me, I look at this as just an expansion of the previous rule where
we have a power to a power because this is a to the first, and this is b to the first. So
if I multiply these exponents, the result would be the same. And the same thing for
the quotient to power rule. This is really a to the first and this is b to the first,
so I look at this and see we have powers raised to powers. Therefore we multiply our exponents.
Okay, we have several examples here. The first one we have seven times x squared. Well,
this is actually seven to the first and x to the first. So applying this product to power rule,
we would have seven squared times x squared, which equals forty-nine x squared. A common
mistake here is that people forget to square the seven as well as the x. The next problem,
I don't see an exponent on the z, so remember that means it's a one.
So here we're going to have x to the two times three power. That
would be x to the sixth, y to the fifth times three power, it's fifteen power,
and then z raised to the one times three or third power. The next example, again,
whenever I don't see an exponent, I always write it in as a one. So I look at this and I see powers
raised to powers. So I would have two to the fourth power divided by three to the fourth power.
And since we can multiply this out, we should. Two to the fourth would give
us sixteen and three to the fourth would be eighty-one. And the last example here,
this would be a to the three times four or twelfth power and b to the two times four or
b to the eighth power. Now that we've discussed the basic properties of exponents,
let's take a look at a couple problems where we will simplify using several of the properties.
So we take a look at this problem. We cannot multiply these two together because two
a squared b is raised to the third power. So we first have to address this outer exponent.
This would be two to the third power, a to the two times three or a to the sixth power,
and this is b to the first. So we'd have b to the third power. Since all
of this is being multiplied together, I'm going to rearrange the order.
Let's write three times two cubed times a factor of a times a to the sixth times b to
the fourth times b cubed. Again, I'm allowed to do this because this is all connected by
multiplication. So now we can multiply this together. Two to the third would be eight.
Eight times three would be twenty-four. a to the first times a to the sixth would give us a to the
seventh since we add our exponents and b to the fourth times b cubed gives us b to the seventh.
Okay, the last example we have a fraction inside these parentheses being raised to
the third power. Now, there's a couple ways to do this problem. We could apply the power rule here,
but I did notice that inside the parentheses, this fraction can be simplified. Here we
have x squared and here we have x to the first. So if we applied the quotient rule,
we'd have x to the two minus one power, which would give us x in the numerator.
And this is still raised to the third power. So if I lost you there, what happened is we had x to the
two in the numerator, and we had one factor of x in the denominator. So when we simplified this,
we were left with x in the numerator, which is the same thing you would get when you
apply the quotient rule where you have x to the two minus one, which is x to the first.
Okay, now we can apply the power rule here, so we'd have x to the one times three power,
x to the third, y to the four times three power, which should be the twelfth power,
two to the one times three or two to the third power, and z to the one
times three would give us z to the third. Two to the third would simplify to eight,
so let's rewrite this one more time: x cubed, y to the twelfth divided by eight z cubed.
Okay, the next video we'll discuss negative exponents. There will be additional examples
of these properties in that video as well. Thank you for watching, and I hope you found it helpful.