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We're asked to simplify r to the 2/3 s to the third, that

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whole thing squared.

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Times the square root of 20r to the fourth s to the fifth.

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Now this looks kind of daunting, but I think if we

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take it step by step it shouldn't be too bad.

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So first we can look at this first expression right here

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where we're taking this product to the second power.

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We know that instead we can take each of the terms in the

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product to the second power and then take the product.

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So this is going to be the same thing as r to the 2/3

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squared times s to the third squared.

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And now let's look at this radical over here.

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We have the square root, but that's the exact same thing as

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raising something to the 1/2 power.

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So this is equal to-- so times this part.

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Let me do this in a different color.

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This part right here, that is the same thing as 20.

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And instead of just writing 20, let me write 20 as the

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product of a perfect square and a non-perfect square.

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So 20 is the same thing as 4 times 5.

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That's the 20 part.

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Times r to the fourth times s to the fifth.

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Now let me write s to the fifth also as a product of a

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perfect square and a non-perfect square.

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r to the fourth is obviously a perfect square.

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Its square root is r squared.

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But let's write s to the fifth in a similar way.

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So s to the fifth we can rewrite as s to the

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fourth times s.

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Right?

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S to the fourth times s to the first, that is s to the fifth.

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And of course, all of this has to be raised to the 1/2 power.

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Now let's simplify this even more.

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If we're taking something to the 2/3 power and then to the

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second power, we can just multiply the exponents.

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So this term right here, we can simplify this as

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r to the 4/3 power.

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And just as a bit of review, taking something to the 4/3

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power, you can view it as either taking-- finding its

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cube root, taking it to the 1/3 power, and then taking its

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cube root to the fourth power.

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Or you can view it as taking it to the fourth power and

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then finding the cube root of that.

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Those are both legitimate ways of something being raised to

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the 4/3 power.

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So you have r to the 4/3 times s to the 3 times 2.

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Times s to the sixth power.

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And then we could raise each of these terms right here to

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the 1/2 power.

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So times-- let me color code it a little bit.

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And we actually wouldn't need the

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parentheses once we do that.

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Times 4 to the 1/2 times 5 to the 1/2.

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That term right there.

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Times r to the fourth to the 1/2 power.

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Times-- I might run out of colors-- s to the fourth to

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the 1/2 power.

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We're raising each of these terms to that 1/2 power.

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Times s to the 1/2 power.

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There's a lot of ways we can go with this, but the one

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thing that might jump out is that there are some perfect

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squares here and we're raising them to the 1/2 power.

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We're taking their square roots, so

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let's simplify those.

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So this 4 to the 1/2, that's the same thing as 2.

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We're taking the principal root of 4.

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5 to the 1/2?

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Well, we can't take the square root of that, so let's just

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write that as the square root of 5.

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r to the fourth to the 1/2.

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There's two ways you can think about it.

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4 times 1/2 is 2.

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So this is r squared.

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Or you could say the square root of r to

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the fourth is r squared.

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So this is r squared.

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Similarly, the square root of s to the fourth or s to the

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1/2 is also s squared.

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And then this s to the 1/2, let's just write that as the

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square root of s.

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Just like that.

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Let's see what else we can do here.

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Let me write these other terms. We have an r to the 4/3

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times s to the sixth times 2 times square root of 5 times r

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squared times s squared times the square root of s.

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Now, a couple of things we can do here.

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We could combine these s terms. Let's do that.

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Actually, just write the 2 out front first. So let's write

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the 2 out front first. So you have 2 times.

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Now let's look at these two s terms over here.

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We have s to the sixth times s squared.

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When someone says to simplify it, there's multiple

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interpretations for it.

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But we'll just say s to the sixth times s squared.

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That's s to the eighth.

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6 plus 2.

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Times s to the eighth power.

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Times-- now this one's interesting and we might want

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to break it up depending on what we consider to be truly

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simplified.

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We have r to the 4/3 times r squared.

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r to the 4/3 is the same thing as r to the 1 and 1/3.

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That's what 4/3 is.

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So 1 and 1/3 plus 2 is 3 and 1/3.

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So we could write this times r to the 3 and 1/3.

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That's a little inconsistent.

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Over here I'm adding a fraction.

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Over here with the s I kind of left out the s to the 1/2 from

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the s's here.

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But we could play around with it and all of those would be

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valid expressions.

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So we've already dealt with the 2.

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We've already dealt with these two s's.

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We've already dealt with these r's.

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And then you have the square root of 5 times the

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square root of s.

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And we could merge them if we want, but I

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won't do it just yet.

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Times the square root of 5 times the square root of s.

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Now there's two ways we could do it.

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We might not like having a fractional exponent here.

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And then we could break it out.

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Or we might want to take this guy and merge it with the

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eighth power.

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Because you know that this is the same

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thing as s to the 1/2.

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So let's do it both ways.

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So if we wanted to merge all of the exponents, we could

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write this as 2 times s to the eighth times s to the 1/2.

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So s to the eighth and s to the 1/2.

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That would be 2 times s to the 8-- I can even

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write it as a decimal.

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8.5.

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8 plus-- you could imagine this is s to the 0.5 power.

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So that's 8.5 times r to the 3 and 1/3.

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I'm kind of mixing notations here.

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I have just a decimal notation, then I have a

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fraction notation, mixed number notation.

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Times the square root of 5.

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This is one simplification.

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I kind of have it in the fewest terms possible.

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The other simplification if you don't want to have these

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fractional exponents out here, you could write it as-- I'll

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do this in a different color.

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You could write this-- and these are all equivalent

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statements.

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So it's up to debate what simplified really means.

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So you could write this as 2 times s to the eighth.

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Instead of writing r to the 3 and 1/3, we could write r to

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the third times the cube root of r, which is the same thing

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as r to the 1/3.

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We could write r to the third times r to the 1/3.

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r to the 1/3 is the same thing as the cube root of r.

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And then you have the square root of these two guys.

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Both of these guys are being raised to the 1/2 power.

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So you could then say times the square root of 5s.

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I like this one a little bit more, the one on the left.

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To me this is really simplified.

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We've merged all of the bases.

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We have these two numbers, we've merged all the s terms,

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all the r terms. This is a little bit more complicated.

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You have a cube root.

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You haven't separated the s's and the r's.

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So I would go with this one if someone really wanted me--

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said hey, Sal, simplify it how you like.

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