1 00:00:01,067 --> 00:00:05,933 simplify 2 a square over 3 b to the fifth 2 00:00:05,933 --> 00:00:08,133 All of that to the fourth power 3 00:00:08,133 --> 00:00:13,933 So here we could use the property that if i have x over y to the Nth power 4 00:00:13,933 --> 00:00:18,000 this is the same thing as x to the Nth power 5 00:00:18,000 --> 00:00:21,667 over y to the nth power. we can see that with an exemple really fast. 6 00:00:21,667 --> 00:00:26,773 if i had x over y to the 3rd power 7 00:00:26,773 --> 00:00:28,333 what would be this equal to? 8 00:00:28,333 --> 00:00:30,600 that would be x over y 9 00:00:30,600 --> 00:00:34,333 times x over y times x over y 10 00:00:34,333 --> 00:00:41,533 which is equal to x times x times x over y times y times y 11 00:00:41,533 --> 00:00:46,000 This is clearly x to the power of 3 over y to the power of 3 12 00:00:46,000 --> 00:00:51,600 particular example of why, hope this makes you feel more confident about it. 13 00:00:51,600 --> 00:00:53,867 Makes you feel good about using this property. 14 00:00:53,867 --> 00:00:55,267 Lets use it right 15 00:00:55,267 --> 00:00:56,733 here in the actual problem 16 00:00:56,733 --> 00:00:59,281 So we have 2 a squared over b to the fifth 17 00:00:59,281 --> 00:01:00,733 all of that to the fourth power 18 00:01:00,733 --> 00:01:02,531 So this going to be the same thing 19 00:01:02,531 --> 00:01:05,933 as to a squared to the fourth power 20 00:01:05,933 --> 00:01:07,867 Let me write it this way. It's going to be 21 00:01:07,867 --> 00:01:13,000 2 a squared to the fourth power, let me do the fourth power in blue 22 00:01:13,000 --> 00:01:16,463 to the fourth power, all of that, 23 00:01:16,463 --> 00:01:20,600 over 3 b to the fifth, all of that over 24 00:01:20,600 --> 00:01:25,800 3 b to the fifth, to the fourth power 25 00:01:25,800 --> 00:01:28,533 to the fourth power 26 00:01:28,533 --> 00:01:31,867 and now we can use the property that if I take the product of things 27 00:01:31,867 --> 00:01:35,086 and raise that to a power, that's the same thing as 28 00:01:35,086 --> 00:01:39,467 raising each of the things in the product to the power 29 00:01:39,467 --> 00:01:42,067 and then taking the product, so this is going to be the same thing 30 00:01:42,067 --> 00:01:43,800 Let me just write the property 31 00:01:43,800 --> 00:01:50,800 So if I have x times y to the nth power, this is the same thing 32 00:01:50,800 --> 00:01:53,333 as x to the nth times y to the nth power. 33 00:01:53,333 --> 00:01:58,538 And you could a similar argument that we did rigth here 34 00:01:58,538 --> 00:02:00,400 to see why this works out. So let's do that over here. 35 00:02:00,400 --> 00:02:04,267 2 a squared to fourth power is going to be equal to 36 00:02:04,267 --> 00:02:08,733 2 to the fourth power, colour coded, 37 00:02:08,733 --> 00:02:12,800 2 to the fourth power times a squared to the fourth power 38 00:02:12,800 --> 00:02:16,667 times a squared to the fourth power 39 00:02:16,667 --> 00:02:18,800 and then all of that over 40 00:02:18,800 --> 00:02:22,467 3 b to the fifth to the fourth power, that's going to be 41 00:02:22,467 --> 00:02:27,133 3 to the fourth power, 3 to the fourth power 42 00:02:27,133 --> 00:02:31,400 times b to the fifth, times b to the fifth, 43 00:02:31,400 --> 00:02:36,067 to the fourth power, times b to the fifth to the fourth power 44 00:02:36,067 --> 00:02:38,400 Now the last interesting thing here, 45 00:02:38,400 --> 00:02:40,733 We can evaluate what this numbers come out to be 46 00:02:40,733 --> 00:02:43,067 2 to the fourth power, let me just do it rigth now 47 00:02:43,067 --> 00:02:47,133 2 to the fourth power, 2 squared is 4, 48 00:02:47,133 --> 00:02:51,667 2 to the third power is 8, 2 to the fourth power is 16 49 00:02:51,667 --> 00:02:53,867 So this rigth here is 16. 50 00:02:53,867 --> 00:02:57,533 3 to the first power is 3, 3 squared is 9, 51 00:02:57,533 --> 00:03:01,533 3 to the third power is 27, 3 to the fourth power is 81. 52 00:03:01,533 --> 00:03:04,867 So this over here, this over here is 53 00:03:04,867 --> 00:03:08,533 let me do it in that green colour, this is 81. 54 00:03:08,533 --> 00:03:12,200 81, and to simplify this terms here 55 00:03:12,200 --> 00:03:13,733 we just have to remember 56 00:03:13,733 --> 00:03:17,067 the what we call to power property or the power rule of exponents. 57 00:03:17,067 --> 00:03:19,200 That's just the notion that if I have, 58 00:03:19,200 --> 00:03:24,267 if I have x to the nth and I raise that 59 00:03:24,267 --> 00:03:27,800 to the mth power, this is the exact same thing as 60 00:03:27,800 --> 00:03:31,400 x to n times m power 61 00:03:31,400 --> 00:03:33,533 So if we want to simplify this, 62 00:03:33,533 --> 00:03:37,400 or numerator is now, it is now 16 63 00:03:37,400 --> 00:03:40,200 that's the 2 to the fourth, and then a squared 64 00:03:40,200 --> 00:03:42,667 and then that to the fourth power, 65 00:03:42,667 --> 00:03:45,533 That's going to be a to the 2 times 4 power, 66 00:03:45,533 --> 00:03:48,467 or a to the 8th power 67 00:03:48,467 --> 00:03:52,267 This is literally 2 times 4, so we are just multiplying 2 times 4 68 00:03:52,267 --> 00:03:56,067 This 2 times this 4 to get that 8. 69 00:03:56,067 --> 00:03:58,399 And then in the denominator 70 00:03:58,399 --> 00:04:03,800 We have our 81, and then b to the fifth and that to the fourth power 71 00:04:03,800 --> 00:04:06,467 That's going to be b to the 5 times 4 power 72 00:04:06,467 --> 00:04:12,935 or b to the 20th power. So I got 20 by multiplying 5 times 4. 73 00:04:12,935 --> 99:59:59,999 And we are done, this is about simplify as we can get.