1
00:00:00,906 --> 00:00:02,396
Rewrite the radical expression

2
00:00:02,396 --> 00:00:05,921
using rational exponents and simplify.

3
00:00:05,921 --> 00:00:09,915
So here we have the fourth root of 5 a to the fourth

4
00:00:09,915 --> 00:00:11,587
b to the twelfth power.

5
00:00:11,587 --> 00:00:16,091
The key thing to realize here is the fourth root of something

6
00:00:16,091 --> 00:00:19,853
is same thing as something to the one fourth power.

7
00:00:19,853 --> 00:00:23,475
Or in particular, or in general, the nth root something

8
00:00:23,475 --> 00:00:27,562
is the same thing as that something to the 1 over n power.

9
00:00:27,562 --> 00:00:29,745
So we can just apply that over here:

10
00:00:29,745 --> 00:00:30,665
the fourth root of all of this

11
00:00:30,665 --> 00:00:35,133
is equal to 5 a to the 4th, b to the 12th power,

12
00:00:35,133 --> 00:00:38,166
all of that to the one fourth power.

13
00:00:38,166 --> 00:00:41,215
And then we also know if we take the product of things,

14
00:00:41,215 --> 00:00:43,491
and then raise them to some exponent,

15
00:00:43,491 --> 00:00:46,928
that's the same thing as raising each of the terms in the product

16
00:00:46,928 --> 00:00:47,998
to the exponent first,

17
00:00:47,998 --> 00:00:50,550
or each of the things that we're taking the product of

18
00:00:50,550 --> 00:00:52,825
to that exponent, and then multiplying.

19
00:00:52,825 --> 00:00:55,057
So let's do that.

20
00:00:55,057 --> 00:00:57,302
So 5, so this is the same thing as

21
00:00:57,302 --> 00:00:59,039
5 to the 1/4 power

22
00:00:59,039 --> 00:01:02,430
times a to the fourth to the 1/4 power

23
00:01:02,430 --> 00:01:07,222
times b to the 12th to the 1/4 power.

24
00:01:07,222 --> 00:01:10,147
Now 5 to the 1/4,

25
00:01:10,147 --> 00:01:11,329
I don't know what that is,

26
00:01:11,329 --> 00:01:13,605
so I'll just keep that as the cube root,

27
00:01:13,605 --> 00:01:15,805
well, we could leave it as 5 to the 1/4,

28
00:01:15,805 --> 00:01:17,876
and that's not not simplified.

29
00:01:17,876 --> 00:01:23,027
or we can just rewrite it again as the fourth root of 5.

30
00:01:23,058 --> 00:01:25,446
a to the fourth to the 1/4 power:

31
00:01:25,446 --> 00:01:28,740
if your raise something to a power and then another power,

32
00:01:28,740 --> 00:01:30,151
and raise that to another power,

33
00:01:30,151 --> 00:01:35,267
that's equivalent of raising a to the four times 1/4 power.

34
00:01:35,267 --> 00:01:36,329
So let me just write that down.

35
00:01:36,329 --> 00:01:41,200
This is, so times a to four times 1/4 power.

36
00:01:41,200 --> 00:01:44,234
And then finally, this right over here,

37
00:01:44,234 --> 00:01:46,928
using the same exact exponent property,

38
00:01:46,928 --> 00:01:50,504
this is b to the 12th times 1/4 power.

39
00:01:50,504 --> 00:01:53,151
So all of this simplifies to,

40
00:01:53,151 --> 00:01:55,241
and I'll change the order here,

41
00:01:55,241 --> 00:01:59,112
so you have the fourth root of 5,

42
00:01:59,112 --> 00:02:02,903
and then you have a to the fourth times 1/4 power

43
00:02:02,903 --> 00:02:05,368
so that's just, this simplifies to a to the first power

44
00:02:05,368 --> 00:02:07,059
which is really just the same thing as a.

45
00:02:07,075 --> 00:02:08,902
So that's just a.

46
00:02:08,902 --> 00:02:13,882
And then we have b to the 12 times 1/4 power.

47
00:02:13,882 --> 00:02:16,454
Well, 12 times 1/4 is just three.

48
00:02:16,454 --> 00:02:19,036
So that's b to the third power.

49
00:02:19,036 --> 99:59:59,999
So it's a, b to the third power times the fourth root of 5.