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Simply 4a squared b and this whole quantity is being raised to the third power

2
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Now here we'll just use the property if we have the product of things,

3
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let's say I have a times b and I'm raising that to some power n

4
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this is just going to be equal to a to the n times b to the n.

5
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And to realize why that happens, let's just try it out with a simple example.

6
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If I have a times b to the third power, what is that going to be equal to?

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That is going to be equal to a times be (that's to the first power)

8
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and then times a times b (and that's to the second power)

9
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times a times b (that's to the third power).

10
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We're literally taking this expression and multiplying it three times.

11
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Now, we can just swap these around, because of the communicative

12
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and associative properties of multiplication.

13
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So if we just swap these around and change the order in which we're multiplying,

14
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we can write this as a times a times a times b times b times b.

15
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We have three a's here and three b's.

16
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So this right here is going to be equal to a to the third,

17
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and this right over here is going to be equal to be to the third.

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So we have a to the third and b to the third, and so it worked.

19
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Hopefully this gives you a sense of why this property holds.

20
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Now let's apply it to the actual problem!

21
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So we have 4 a squared b to the third power.

22
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That means that each of the components in the product are going to b raised

23
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to the third power so that means that we have 4 to the third power times

24
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a squared to the third power times b to the third power

25
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let me color code, this third power

26
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here in mangenta, that's this third power over here

27
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I'm just raising it, sort of taking the product first and then

28
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raising that to the third power

29
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I can take each of the terms in the product eah of the numbers

30
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in the product and raised those to third power first

31
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and then take the product. So that's what we are doing

32
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Let's try to simplify this

33
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What is four, what is four to the third power

34
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Four to the first is four, four to the second is 16

35
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and we want to multiply 16 times 4,

36
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So 4 to the third power is just 4 times 4 times 4 is 64.

37
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and then we have a squared and we are raising that

38
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to the third power. We know from the product property of

39
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the exponents that's going to be a to the 2 times 3 power

40
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or a to the sixth power, so this is a to the sixth power

41
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we got six from 2 times three and then finaly

42
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we juste have b to the third power, let me do that in a different colour

43
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we just have b to the third power and I'll juste write it over here

44
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So we simplify this expression to 64 a to the sixth

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b to the third power.