The German Math Olympiad Problem That 99% of Students Get Wrong!

Added: 2026-05-11

[00:00:00]You're welcome to solve this nice math problem, which is 64 to the power of m.

[00:00:06]This is equal to the square root of 16 multiplying by square root of 8.

[00:00:10]So, what is the value of m?

[00:00:12]Now, let's provide a solution from here.

[00:00:18]So, we are going to solve this math problem by applying two methods.

[00:00:22]Let's start with method one.

[00:00:26]Method one.

[00:00:28]Now, in method one, we have 64 raised to the power of m.

[00:00:33]This is equal to square root of 16 multiplying by square root of 8.

[00:00:41]So, the first step is to square on both sides. Let's square on both sides.

[00:00:47]Now, we square on both sides here so that now here we have 64 raised to the power of m raised to the power of two.

[00:00:58]This is equal to Let's eliminate the first square root sign so that you have 16 multiplying by square root of 8.

[00:01:09]Now, the next step is that 64 to the power of m raised to the power of two.

[00:01:13]This is in the form of a to the power of n raised to the power of m, which we can express as a to the power of n multiplying by m.

[00:01:23]So, this implies that here we have 64 raised to the power of two m.

[00:01:29]This is equal to 16 multiplying by square root of 8.

[00:01:36]The next step is to square again on both sides.

[00:01:40]Let's square on both sides so that now we have 64 raised to the power of 2m * 2. This is 4m.

[00:01:51]And this is equal to 16 raised to the power of two multiplying by square root of eight raised to the power of two.

[00:02:04]So, we have 64 raised to the power of 4m.

[00:02:09]This is equal to 16 raised to the power of two multiplying by Now, let's eliminate the second square root sign. So, this is multiplying by eight.

[00:02:20]Now, we can express 64. This is the same thing as two to the power of six. This is raised to the power of 4m.

[00:02:28]And this is equal to 16, which is two to the power of four. This is raised to the power of two multiplying by eight, which is two to the power of three.

[00:02:40]Now, the next step is that we have two raised to the power of six times 4m. So, this means we have two to the power of 24m.

[00:02:50]This is equal to two raised to the power of four times two, which is eight multiplying by two raised to the power of three.

[00:03:00]Okay.

[00:03:01]Now, we have that two to the power of eight times two to the power of three.

[00:03:05]This is in the form of a to the power of n times a to the power of m, which we can express as a to the power of n plus m.

[00:03:14]So, this implies that we have two to the power of 24 m.

[00:03:20]This is equal to two raised to the power of eight plus three.

[00:03:26]So, we have two to the power of 24m.

[00:03:30]This is equal to raised to the power of eight plus three. This is equal to 11.

[00:03:36]And since the bases are common here, that is a to the power of n, this is equal to a to the power of m. Since the bases are common, exponent n is equal to m.

[00:03:49]So, this is to mean that a we have 24 m this is equal to 11.

[00:03:57]So, let's divide on both sides by 24.

[00:04:01]So, that now we have the value of m >> [clears throat] >> is equal to 11 over 24.

[00:04:08]So, this is the value of m.

[00:04:11]So, let's apply method two. Let's apply the second method.

[00:04:17]Method two.

[00:04:19]So, we have 64 raised to the power of m this is equal to square root of 16 multiplying by square root of eight.

[00:04:33]We have that a square root of a this can also be expressed as a raised to the power of a half.

[00:04:42]So, this is to mean that, yeah, we have 64 raised to the power of m this is equal to 16 raised to the power of a half multiplying by eight raised to the Remember, eight is under two square root sign here. So, this is eight raised to the power of a half then raised to the power of a half.

[00:05:06]Okay?

[00:05:09]Now, the next step is that we have 64 raised to the power of m this is equal to 16 We can express 16 as two raised to the power of four then raised to the power of a half multiplying by eight Eight is the same thing as two raised to the power of three this is raised to the power of a half then raised to the power of a half.

[00:05:37]This is to mean that this is in the form of a to the power of n raised to the power of m which we can express as a to the power of n multiplied by m.

[00:05:48]So, we have 64 which is the same thing as 2 to the power of 6.

[00:05:54]Then, raised to the power of m. This is equal to 2 raised to the power of 4 times 1/2 then multiplying by 2 raised to the power of 3. 1/2 * 1/2 is 1/4. So, this is 3 * 1/4.

[00:06:13]So, this is 2 raised to the power of 6 * m which is 6m. This is equal to raised to the power of 4 / 2 is 2. So, this is 2 raised to the power of 2 * 2 to the power of 3 * 1/4. So, this is 2 to the power of 3/4.

[00:06:32]So, we have 2 to the power of 6m.

[00:06:36]This is equal to 2 raised to the power of 2 plus 3/4.

[00:06:43]We have that 2 plus 3/4.

[00:06:47]The LCM is 4.

[00:06:50]4 / 1 is 4 * 2 which is 8 + 4 / 4 is 1 * 3. This is 3. So, here we have 8 + 3 which is 11/4.

[00:07:01]So, this is to mean we have 2 to the power of 6m.

[00:07:05]This is equal to 2 raised to the power of 11/4.

[00:07:11]Since the bases are common here then we have 6m this is equal to 11/4.

[00:07:20]So, let's multiply both sides by 1/6.

[00:07:24]Multiplying by 1/ 6.

[00:07:27]So, that now we have m is equal to 11 over 6 * 4 which is 24.

[00:07:37]So, this is the value of m by applying method two. So, both method one and method two yields the same value of m which is 11 over 24.

[00:07:49]So, let's verify this value of m satisfies the equation.

[00:07:54]Now, let's verify that this value of m, which is 11 over 24, if this satisfies the equation.

[00:08:01]Now, if you recall, we have 64 raised to the power of m.

[00:08:07]This should be equal to square root of 16 multiplying by square root of eight.

[00:08:16]So, let's substitute the value of m, which is 11 over 24.

[00:08:20]So, this is 64 raised to the power of 11 over 24. This is should be equal to 16 raised to the power of a half.

[00:08:32]This can be expressed as 16 raised to the power of a half times eight raised to the power of a half then raised to the power of a half.

[00:08:43]So, 64 is the same thing as two to the power of six raised to the power of 11 over 24.

[00:08:52]This should be equal to 16, which is two to the power of four raised to the power of a half times eight, which is two to the power of three raised to the power of a half times a half, which is one over four.

[00:09:08]So, this means we have two to the power of six multiplying by 11 over 24.

[00:09:16]This should be equal to two raised to the power of four times a half, which is two to the power of two then times two raised to the power of three times one over four. This is three over four.

[00:09:31]Again, we have that two to the power of two times two to the power of three over four. This is in the form of a to the power of n times a to the power of m which we can express as a to the power of n plus m.

[00:09:45]So this means that two plus three over four This is the same thing as four divided by one is four times two this is eight plus four divided by four is one times three this is plus three.

[00:09:58]And this is equal to eight plus three which is eleven over four.

[00:10:03]So this depict that we have two to the power of six times eleven over four.

[00:10:10]This should be equal to two raised to the power of eleven over four.

[00:10:17]Now let's simplify here six divided by two.

[00:10:21]If you see Here we have two to the power of six times eleven over twenty-four.

[00:10:26]So that we have six divided six divided by six is one twenty-four divided by six this is equal to four.

[00:10:34]So we have two raised to the power of eleven over four this is equal to two raised to the power of eleven over four.

[00:10:43]So we have that the left hand side is equal to the right hand side and this affirms that the value of m which is eleven over twenty-four satisfies the equation.

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[00:11:04]See you in the next video.