Hey everyone, it's Justin again. In the last three videos, you learned the
different rules for simplifying exponents. Because there was so much, in this video,
we'll review all the rules and then do some examples that will mix them together.
First, let's review those exponent rules. The product rule says that if you multiply powers
with the same base, the exponents will add together. The quotient rule is very similar,
except when you divide powers with the same base, you subtract the exponents. The power
of a product rule says what to do if you've got a product of two values all being raised to a power:
spread the power to all the multipliers. Remember, this only works with multiplication
inside the parentheses, so don't go trying this when there's addition or subtraction.
The power of a power rule tells you that if a power is raised to another power,
the two exponents will multiply together. The power of a quotient rule is like the power of
a product rule except with fractions. The outer exponent spreads to both the numerator and the
denominator. Finally, negative exponents means there's a flip. The power is in the
wrong place and needs to move down if it's on the top, or up if it's on the bottom.
Now that we've reviewed the rules, let's practice. First, here we've got products being raised to
power, so we can use the power of a product rule to distribute the exponents. 5 and x squared get
the exponent of 2, and on the bottom 5 and x get exponents of -1. We've got a power inside another
power, so let's use the power of a power rule. Since there are no operations inside the other
parentheses, we'll remove those while we're at it. Now, we've got a few bases that appear both on top
and bottom, so we can use the quotient rule. For a base of 5, we'll subtract the exponents 2 and -1,
which gives us 5 to the third power. For the base of x, 4 minus -1 results in x to the fifth power.
All we have left to do now is simplify. 5 to the third power, 5 times 5 times 5, is 125
times x to the fifth power. And there we have it, our final simplified answer.
For the next example, we'll start by simplifying the fraction using the quotient rule. For a,
2 minus 3 is -1, and for b, 3 minus 2 is 1. Now we've got powers being multiplied, so let's use
the product rule. a to the negative first power times a to the first power,
we'll add the exponents and make 0, while the b's combine to make b squared.
At this point, we'll be using a special property of exponents.
Anything to the zeroth power is automatically 1. So, a to the zeroth power can be rewritten
as 1. Finally, anything times 1 just stays the same, so 1 times b squared is just b squared.
Here's one last example. It's a doozy. You're gonna need nearly every rule in
order to simplify this one. I challenge you to pause and see if you can do it yourself.
Did you try it on your own? If so, let's see how you did.
First, let's use power of a quotient and power of a product to spread the -2 to the numerator
and the denominator, then spread it further to each piece. These two rules are often combined
into one step because of how similar they are. Is it just me or does this look like a hot mess?
Let's condense some of these exponents using power of a power, which says to multiply the exponents.
Now, this still isn't very clean, so let's flip all those negative exponents to where they go.
2 squared and q to the 8th power belong on the bottom, and 3 squared and r to
the -4th power belong on top. There we go, now it's looking a little cleaner.
Let's move that up so we have some more room. Now, we can multiply these expressions together,
and while we're at it, let's simplify 3 squared on top and 2 squared on the
bottom. We're getting closer to the final result, but we've got a couple problems here.
First, two powers with the same base being multiplied. For that, we'll use the product
rule and add the exponents together to make s to the 11th power. The last thing we need
to simplify are the q's in the numerator and the denominator. Since we're dividing them,
we'll use the quotient rule. If we subtract the exponents 8 minus 8 will get 0, which means
there are no more q's. Remember q to the zeroth power is the same as 1. If we multiply by 1,
it won't do anything, so it just becomes invisible. This is now the most simplified
version of our final answer. You might have used some of the rules in different orders,
and that's totally okay. As long as you've got this answer, you did it correctly.
Exponents add a new layer of possibilities to math, but also more complexity. Practice makes
perfect, so keep it up! Up next, we'll begin learning how to use,
manipulate, and simplify the inverse of exponents: roots. See you then!