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# Simplifying mixed radical and exponential expressions

Video transcript

We're asked to simplify r to the
2/3 s to the third, that whole thing squared. Times the square root of 20r to
the fourth s to the fifth. Now this looks kind of daunting,
but I think if we take it step by step it
shouldn't be too bad. So first we can look at this
first expression right here where we're taking this product
to the second power. We know that instead we can take
each of the terms in the product to the second power
and then take the product. So this is going to be the same
thing as r to the 2/3 squared times s to the
third squared. And now let's look at this
radical over here. We have the square root, but
that's the exact same thing as raising something to
the 1/2 power. So this is equal to--
so times this part. Let me do this in a
different color. This part right here, that
is the same thing as 20. And instead of just writing
20, let me write 20 as the product of a perfect square
and a non-perfect square. So 20 is the same thing
as 4 times 5. That's the 20 part. Times r to the fourth times
s to the fifth. Now let me write s to the fifth
also as a product of a perfect square and a
non-perfect square. r to the fourth is obviously
a perfect square. Its square root is r squared. But let's write s to the
fifth in a similar way. So s to the fifth we can
rewrite as s to the fourth times s. Right? S to the fourth times s to the
first, that is s to the fifth. And of course, all of this has
to be raised to the 1/2 power. Now let's simplify
this even more. If we're taking something to the
2/3 power and then to the second power, we can just
multiply the exponents. So this term right here,
we can simplify this as r to the 4/3 power. And just as a bit of review,
taking something to the 4/3 power, you can view it as either
taking-- finding its cube root, taking it to the 1/3
power, and then taking its cube root to the fourth power. Or you can view it as taking
it to the fourth power and then finding the cube
root of that. Those are both legitimate ways
of something being raised to the 4/3 power. So you have r to the 4/3 times
s to the 3 times 2. Times s to the sixth power. And then we could raise each of
these terms right here to the 1/2 power. So times-- let me color
code it a little bit. And we actually wouldn't
need the parentheses once we do that. Times 4 to the 1/2 times
5 to the 1/2. That term right there. Times r to the fourth
to the 1/2 power. Times-- I might run out of
colors-- s to the fourth to the 1/2 power. We're raising each of these
terms to that 1/2 power. Times s to the 1/2 power. There's a lot of ways we can
go with this, but the one thing that might jump out is
that there are some perfect squares here and we're raising
them to the 1/2 power. We're taking their
square roots, so let's simplify those. So this 4 to the 1/2, that's
the same thing as 2. We're taking the principal
root of 4. 5 to the 1/2? Well, we can't take the square
root of that, so let's just write that as the square
root of 5. r to the fourth to the 1/2. There's two ways you
can think about it. 4 times 1/2 is 2. So this is r squared. Or you could say the
square root of r to the fourth is r squared. So this is r squared. Similarly, the square root of
s to the fourth or s to the 1/2 is also s squared. And then this s to the 1/2,
let's just write that as the square root of s. Just like that. Let's see what else
we can do here. Let me write these other terms.
We have an r to the 4/3 times s to the sixth times 2
times square root of 5 times r squared times s squared times
the square root of s. Now, a couple of things
we can do here. We could combine these s
terms. Let's do that. Actually, just write the 2 out
front first. So let's write the 2 out front first.
So you have 2 times. Now let's look at these
two s terms over here. We have s to the sixth
times s squared. When someone says to simplify
it, there's multiple interpretations for it. But we'll just say s to the
sixth times s squared. That's s to the eighth. 6 plus 2. Times s to the eighth power. Times-- now this one's
interesting and we might want to break it up depending on what
we consider to be truly simplified. We have r to the 4/3
times r squared. r to the 4/3 is the same thing
as r to the 1 and 1/3. That's what 4/3 is. So 1 and 1/3 plus
2 is 3 and 1/3. So we could write this times
r to the 3 and 1/3. That's a little inconsistent. Over here I'm adding
a fraction. Over here with the s I kind of
left out the s to the 1/2 from the s's here. But we could play around with
it and all of those would be valid expressions. So we've already dealt
with the 2. We've already dealt with
these two s's. We've already dealt
with these r's. And then you have the square
root of 5 times the square root of s. And we could merge them
if we want, but I won't do it just yet. Times the square root of 5 times
the square root of s. Now there's two ways
we could do it. We might not like having a
fractional exponent here. And then we could
break it out. Or we might want to take this
guy and merge it with the eighth power. Because you know that
this is the same thing as s to the 1/2. So let's do it both ways. So if we wanted to merge all
of the exponents, we could write this as 2 times s to the
eighth times s to the 1/2. So s to the eighth
and s to the 1/2. That would be 2 times s
to the 8-- I can even write it as a decimal. 8.5. 8 plus-- you could imagine this
is s to the 0.5 power. So that's 8.5 times r
to the 3 and 1/3. I'm kind of mixing
notations here. I have just a decimal notation,
then I have a fraction notation, mixed
number notation. Times the square root of 5. This is one simplification. I kind of have it in the
fewest terms possible. The other simplification if you
don't want to have these fractional exponents out here,
you could write it as-- I'll do this in a different color. You could write this-- and
these are all equivalent statements. So it's up to debate what
simplified really means. So you could write this as
2 times s to the eighth. Instead of writing r to the 3
and 1/3, we could write r to the third times the cube root of
r, which is the same thing as r to the 1/3. We could write r to the third
times r to the 1/3. r to the 1/3 is the same thing
as the cube root of r. And then you have the square
root of these two guys. Both of these guys are being
raised to the 1/2 power. So you could then say times
the square root of 5s. I like this one a little bit
more, the one on the left. To me this is really
simplified. We've merged all of the bases. We have these two numbers, we've
merged all the s terms, all the r terms. This is a
little bit more complicated. You have a cube root. You haven't separated
the s's and the r's. So I would go with this one if
someone really wanted me-- said hey, Sal, simplify
it how you like.