WEBVTT
00:00:00.480 --> 00:00:09.200
Let π₯ be equal to the fifth root of the cubed root of seven over the fourth root of two.
00:00:10.160 --> 00:00:12.610
Which of the following numbers is rational?
00:00:13.290 --> 00:00:24.840
(A) π₯ cubed, (B) π₯ to the fourth power, (C) π₯ to the 12th power, (D) π₯ to the 15th power, or (E) π₯ to the 60th power.
00:00:25.300 --> 00:00:31.110
Before we start, there are two things we should think about, the first of them being what is a rational number.
00:00:31.960 --> 00:00:42.310
A rational number is a number that can be written in the form π over π, where π and π are integers and π is not equal to zero.
00:00:42.980 --> 00:00:49.510
Another way to say this is that a rational number can be made by dividing two integers.
00:00:50.190 --> 00:00:56.720
Now, the second thing we might want to think about is seeing if we can simplify what π₯ is equal to here.
00:00:57.200 --> 00:01:06.200
To do that, letβs think about our exponent rule that tells us that the πth root of π₯ is equal to π₯ to the one over π power.
00:01:06.710 --> 00:01:12.390
And that means we can rewrite the fifth root as the one-fifth power.
00:01:13.210 --> 00:01:26.310
And we wanna repeat this process with the two roots inside the parentheses, which means weβll have seven to the one-third power over two to the one-fourth power, all taken to the one-fifth power.
00:01:26.940 --> 00:01:38.730
And one further step we can do to simplify is remember the power of a power rule that tells us π₯ to the π power to the π power is equal to π₯ to the π times π power.
00:01:39.150 --> 00:01:43.450
And this means we can multiply one-third by one-fifth.
00:01:43.770 --> 00:01:50.980
When we do that, we multiply the numerators, one times one is one, then multiply the denominators, three times five is 15.
00:01:51.360 --> 00:01:53.470
Weβll have seven to the 15th power.
00:01:53.900 --> 00:01:59.880
And then weβll need to multiply one-fourth times one-fifth to find the power of our denominator.
00:02:00.420 --> 00:02:03.280
One-fourth times one-fifth is one twentieth.
00:02:03.720 --> 00:02:11.140
And so weβre saying that π₯ is equal to seven to the one fifteenth power over two to the one twentieth power.
00:02:11.940 --> 00:02:20.110
To find a rational value, we need the numerator and the denominator here to be an integer.
00:02:20.650 --> 00:02:22.640
So letβs consider some of our options.
00:02:22.960 --> 00:02:29.970
If we cube π₯, weβll have seven to the one fifteenth power over two to the one twentieth power cubed.
00:02:30.540 --> 00:02:35.700
And if we were going to take a power of a power, we would multiply these powers together.
00:02:36.210 --> 00:02:45.730
We would have π₯ cubed being equal to seven to the three fifteenth power over two to the three twentieths power.
00:02:46.490 --> 00:02:50.880
Now, we could plug this value into our calculator to see what would happen.
00:02:51.460 --> 00:02:55.420
When we do that, we get 1.3304 continuing.
00:02:55.900 --> 00:02:58.100
This is an irrational value.
00:02:58.790 --> 00:03:05.130
And so maybe we should consider what kind of powers here would give us integers.
00:03:06.110 --> 00:03:15.340
If our exponents were whole numbers, if we were taking seven to an integer power, the outcome would be an integer.
00:03:16.030 --> 00:03:21.790
And that means we want to take both two and seven to some power of an integer.
00:03:22.330 --> 00:03:30.810
We need to multiply one fifteenth and one twentieth by some number that produces an integer for both of these values.
00:03:31.510 --> 00:03:38.980
For example, if we took π₯ to the 15th power, then we would have seven to the first power.
00:03:39.440 --> 00:03:42.350
But we would still have a fraction in our denominator.
00:03:42.850 --> 00:03:45.240
But that does get us a bit closer.
00:03:45.760 --> 00:03:59.620
From our list, we can take π₯ to the 60th power, which would be seven to the 60 over 15 power and two to the 60 over 20 power.
00:04:00.520 --> 00:04:02.950
60 divided by 15 is four.
00:04:03.600 --> 00:04:06.010
60 divided by 20 is three.
00:04:06.380 --> 00:04:12.630
π₯ to the fourth power will be an integer, and two cubed will also be an integer.
00:04:13.140 --> 00:04:17.000
An integer divided by an integer is a rational number.
00:04:17.530 --> 00:04:23.050
And so we can say that π₯ to the 60th power will be rational.