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We're asked to simplify the cube root of twenty seven a squared times b to the fifth

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times c to the third power.

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And the goal whenever you try to just simplify a cube,a cube root like this, is we wanna look

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at the parts of this exepression

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over here,

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that are perfect cubes that are something raised to the 3rd power

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that we can take just the cube root of those, since you taking them out of the radical sign

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and then leaving everything else that is not a perfect cube inside of it

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so let's see what we can do

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so first of all 27, you may or may not already recognise this

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is a perfect cube,if you don't already recognise it, you can

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actually do a prime factorisation and see it is a perfect cube

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27 is 3 times 9 and

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9 is 3 times 3 so 27

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prime factorisation is 3 times 3 times 3, so is the exact

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same thing as 3 to the third power, so

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let's rewrite this whole expression down here, let's try in terms of things

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that are perfect cube and things that aren't, so 27 can be just rewritten

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as 3 to the third power, then you have

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a squared, clearly not a perfect cube, but if the third would have been

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so we just gonna write this, let me write it over here

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we can switch the order here, cause we just have a bunch of things being multiple by each other, so i'll write

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the a squared over here,b to the fifth

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b to the fifth is not a perfect cube by itself

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it can be, it can be expressed as a product of a perfect cube

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in another thing, b to the fifth is the exact same thing as the

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b to the third power times b to the second power

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if you wanna see that explicitly b to the fifth is b times b times

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b times b times b

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so this, the first three are clearly b to the third power

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and then you have b to the second power after it, so we can rewrite

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b to the fifth as a product of a perfect cube, so

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i'll write b to the third (doing the same purple colour)

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so we have b to the third power over here

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and this b to the thirds times b squared, so i'll write the b squared

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over here, we assume we gonna multiply all this stuff, and then finally

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finally, we have(other than blue)

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c to the third power, clearly this is a perfect cube

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it is c cube,it is c to the third power, so i'll

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put it over here, so this is c to the third power, and of cause we still have

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that over arching radical sign, so we still

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trying to take the cube root of all of this, and we know

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from our exponent properties, or we could say from our radical properties

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that this is the exact same thing, that taking the cube root of all of this

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things is the same as taking the cube root of these individual factors

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and then multiplying them, so this is the same thing as

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the cube root, and I can separate them out individually or I could say that

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cube root of 3 to the third, b to the third

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c to the third,I'll have to use it both ways, so i'll start taking them

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out separately, so this is the same thing as the cube root of

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three to the third times the cube root ( I'll write

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them all it, while I write do it we'll colour code so we all don't get confused )

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times the cube root of to b to the third times

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the cube root times the

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cube root of c to the third

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c to the third, times the cube root

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and i'll just group these two guys together just because we're not gonna be able to

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simply them anymore, times the cube root, times the

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cube root of a squared b squared

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( I'll keep the colour consistent while we are trying to figure out what's what )

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a squared, b squared, now I could have, could have said

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that this is times the cube root of a squared times the cube root of b squared, but that won't

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simplify anything, but that won't simplify anything so i'll just leave these like this, and so we

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can look as these individually the cube root of 3 to the third or the cube root of

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27, well that's clearly just gonna be ( i'm gonna do that in that

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yellow colour ) this is clearly just gonna be 3

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right, 3 to the third power is three to the third power

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or it is equal to 27, this term right over here, the cube root

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of b to the third, well that is just b, that's just b

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and the cube root of c to the third, the cube root of c to the third, well

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that is clearly (won't do that in that ) that is

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clearly just c

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So our whole expression have simplified to

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3 times b times c

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times c, times the cube root

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times the cube root, of a squared b squared

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times the cube root of a squared

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a squared, b squared, and we're done

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i just wanna do just one other thing, just cause I did mentioned I would do it

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we could simplify it this way, or we could recoignise, we could recoignise

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that this expression right over here can be written as

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3 b c

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to the third power, but if I take 3 things to the third power and i'm multiplying it, that is the same thing as

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multiplying them first and then raising to the third power, comes straight out

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of our exponent properties, and so we can rewrite this as

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cube root, the cube root of all of this times

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the cube root, times the cube root of a squared

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b squared, and so the cube root of all this of 3 b

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c of the third power, well that's just gonna be 3

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b c and then multiply it by the cube root of

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a squared b squared, I didn't take the trouble to colour code it this time cause we already

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figured out one way to solve it, but hopefully that also makes sense, we could have gone

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that we could have done this either way, but the important thing is that we get the same answer.