WEBVTT
00:00:02.990 --> 00:00:11.040
Given that eight to the power of π¦ equals four to the power of π§ equals 64, find the value of π¦ plus π§.
00:00:11.040 --> 00:00:16.620
The first stage of solving this problem is gonna be finding π¦.
00:00:17.220 --> 00:00:21.730
So what Iβve done is Iβve actually put eight to the power of π¦ is equal to 64.
00:00:22.830 --> 00:00:25.560
So therefore I can say that π¦ is equal to two.
00:00:26.930 --> 00:00:35.040
And we know that because we know that eight squared is equal to 64 or eight is actually the root of 64.
00:00:35.590 --> 00:00:36.550
Okay, great!
00:00:36.770 --> 00:00:37.810
So now weβve found π¦.
00:00:37.840 --> 00:00:39.210
Letβs move on and find π§.
00:00:40.770 --> 00:00:42.670
There are actually a couple of ways we could find π§.
00:00:43.290 --> 00:00:48.350
What Iβm gonna do is Iβm gonna use one method now and then use another method to actually check our answer.
00:00:50.140 --> 00:00:54.470
So first of all, we have eight to the power of π¦ is equal to four to the power of π§.
00:00:54.470 --> 00:01:10.750
And what Iβm gonna do is Iβm gonna actually use these two relationships to actually make them have the same base number, because what we have is eight is equal to two to the power of three and four is equal to two to the power of two or two squared.
00:01:12.250 --> 00:01:20.240
So therefore, what we actually have is two to the power of three to the power of π¦ is equal to two squared or two to the power of two to the power of π§.
00:01:22.040 --> 00:01:37.530
So what Iβm gonna do now is Iβm actually gonna use one of our exponent rules to actually simplify even further, because what we have is π₯ to the power of π to the power of π is equal to π₯ to the power of ππ.
00:01:37.930 --> 00:01:40.000
So we actually multiply the exponents.
00:01:41.470 --> 00:01:47.570
So therefore, we have two to the power of three π¦ is equal to two to the power of two π§.
00:01:47.570 --> 00:01:51.080
And thatβs because we multiply three and π¦ and two and π§.
00:01:52.950 --> 00:01:59.780
And now because weβve actually got the same base number either side of our equation, we can now equate the exponents.
00:02:01.270 --> 00:02:04.580
So therefore, we have three π¦ is equal to two π§.
00:02:06.430 --> 00:02:11.020
So at this point, what we can do is actually substitute back in our value for π¦.
00:02:11.260 --> 00:02:12.920
And we know that π¦ is equal to two.
00:02:14.620 --> 00:02:17.460
So weβre gonna get three multiplied by two is equal to two π§.
00:02:19.550 --> 00:02:22.740
So therefore, we have a value of π§ which is equal to three.
00:02:23.430 --> 00:02:27.860
And we get that because we have three multiplied by two, so we divide each side by two.
00:02:28.370 --> 00:02:29.460
Then weβre left with three.
00:02:31.940 --> 00:02:36.010
And as I said, we can check that using the other method that we couldβve used to find π§.
00:02:36.010 --> 00:02:41.570
And we have four to the power of π§ is equal to 64.
00:02:43.140 --> 00:02:54.270
So therefore, π§ will be equal to three as we know that four to the power of three is equal to 64 because four multiplied by four is 16 multiplied by another four is 64.
00:02:54.690 --> 00:02:55.410
Okay, great!
00:02:55.410 --> 00:02:57.460
So we found π¦ and π§.
00:02:57.960 --> 00:02:58.850
And weβve checked π§.
00:03:00.960 --> 00:03:05.080
So now we just move on to the final part of the question, which to find π¦ plus π§.
00:03:06.360 --> 00:03:07.340
So weβve got π¦ plus π§.
00:03:07.340 --> 00:03:12.000
And then we substitute our value of π¦ equals two and our value that π§ equals three.
00:03:12.870 --> 00:03:14.270
So we get two plus three.
00:03:15.110 --> 00:03:24.420
So therefore, we can say that given that eight to the power of π¦ equals four to the power of π§ equals 64, then the value of π¦ plus π§ is equal to five.