Germany | A Nice Algebra Problem | Math Olympiad

[00:00:00]You're welcome to solve this nice math problem, which is 9 over 4 raised to the power of 9 over 4. So, let's provide a solution from here.

[00:00:11]Now, this math problem we are going to solve by applying two methods. Let's start with method one.

[00:00:19]Now, in method one, this is 9 over 4 raised to the power of 9 over 4.

[00:00:30]Now, this is equal to we can express 9 over 4. This is the same thing as 3. 9 is 3 squared divided by 4, which is 2 squared, and this is raised to the power of 9 over 4.

[00:00:48]In the next step, we have that 3 squared over 2 squared. This is in the form of a to the power of n over b to the power of n, and this can be expressed as a over b raised to the power of n.

[00:01:04]Now, let's apply this property here so that we have 3 over 2 raised to the power of 2 then raised to the power of 9 over 4.

[00:01:22]So, therefore, here we have 3 over 2 raised to the power of 2 multiplying by 9 over 4.

[00:01:36]So, therefore, we have 3 over 2 raised to the power of Now, let's simplify here. We have 4 divided by 2.

[00:01:46]This is 2, so we have 9 over 2.

[00:01:48]3 over 2 raised to the power of 9 over 2.

[00:01:54]In the next step, we can express 9 over 2. This is the same thing as 3 over 2 raised to the power of 9. This is the same thing as 4 plus 4 plus 1 and this is divided by 2.

[00:02:13]This is divided by 2. So, in this case here we have 3 over 2 raised to the power of 4 over 2 plus 4 over 2 then plus a half.

[00:02:31]In the next step, here we have 3 over 2 raised to the power of 4 over 2 which is the same thing as 2.

[00:02:41]Then plus 4 over 2. This is 2.

[00:02:45]Then plus a half.

[00:02:49]So, therefore, here we have 3 over 2 raised to the power of 2 plus 2. This is 4 plus a half.

[00:03:04]Now, in the next step, we have that 3 over 2 raised to the power 4 plus a half. This is in the form of a raised to the power of n plus m and this is a to the power of n times a to the power of m.

[00:03:19]Now, let's apply this property so that now this is 3 over 2 >> [snorts] >> raised to the power of 4 multiplying by 3 over 2 raised to the power of a half.

[00:03:34]And therefore, here we have 3 over 2 to the power 4. This is the same thing as 3 to the power of 4 divided by 2 to the power of 4 multiplying by 3 over 2 raised to the power of a half. This is the same thing as the square root of three over two.

[00:03:55]So, we have 33 raised to the power of four. This is the same thing as 81 divided [snorts] by two to the power of four, which is 16.

[00:04:06]Then multiplying by We can express the square root of three over two. This is the same thing as the square root of three over square root of two.

[00:04:16]In the next step, let rationalize the denominator here by multiplying the denominator by root two and also the numerator by root two.

[00:04:28]And therefore, we have 81 over 16 multiplying by root three times root two. This is the same thing as root six.

[00:04:39]Then dividing by root two times root two. This is We have root two raised to the power of two.

[00:04:51]Therefore, here we have 81 multiplying by root six divided by We have 16 multiplying by Let's eliminate the square root sign here. Multiplying by two.

[00:05:07]So, we have the solution here as 81 multiplying by square root of six divided by 16 times two, and this is equal to 32.

[00:05:19]So, this is the solution by applying method one.

[00:05:24]So, let's proceed to method two.

[00:05:28]Now, in method two here, we have nine over four raised to the power of nine over four.

[00:05:33]This is the same thing as nine over four raised to the power of nine over four.

[00:05:40]We can express nine over four.

[00:05:43]This is the same thing as nine is the same thing as eight plus one divided by four.

[00:05:51]So therefore we have a This is 8 / 4 + 1 / 4.

[00:05:59]So therefore we have 8 / 4.

[00:06:02]This is the same thing as 2 + 1 / 4. So let's substitute 2 + 1 / 4.

[00:06:09]So therefore we have 9 / 4 raised to the power of 2 + 1 / 4.

[00:06:21]So this is actually in the form of a to the power of n + m, which we can express as a to the power of n times a to the power of m.

[00:06:33]Applying this exponent property, we have 9 / 4 raised to the power of 2 multiplying by 9 / 4 raised to the power of 1 / 4.

[00:06:51]Okay.

[00:06:52]So here we have >> [snorts] >> 9 / 4 to the power of 2. This is the same thing as 9 squared / 4 squared.

[00:07:03]And this is multiplying by 9 / 4.

[00:07:06]9 is same thing as 3 squared / 2 squared.

[00:07:12]This is raised to the power of 1 / 4.

[00:07:17]So therefore here we have 9 squared.

[00:07:21]This is 81 / 4 squared, which is 16, multiplying by 3 / 2 raised to the power of 2 times 1 / 4.

[00:07:34]Okay.

[00:07:35]So this is 81 / 16 multiplying by 3 / 2 raised to the power of Let's simplify here.

[00:07:45]So, 4 / 2, this is 2. So, we have 3 over 2 raised to the power of 1/2. Raised to the power of 1/2. We have 81 over 16 * 3 over 2 raised to the power of 1/2.

[00:07:58]So, therefore here we have 81 over 16 * the square root of 3 over 2.

[00:08:10]Now, here we have 81 over 16 multiplying by square root of 3 over 2.

[00:08:17]This is the same thing as root 3 over root 2.

[00:08:23]So, let's multiply.

[00:08:25]Let's rationalize the denominator here by multiplying by root 2. And also the numerator, let's multiply it by root 2.

[00:08:35]So, here we have 81 over 16.

[00:08:40]This is multiplying by root 3 * root 2. This is root 6.

[00:08:46]Dividing by root 2 * root 2, this is 2.

[00:08:52]And therefore, we have the solution here, which is 81 root 6.

[00:08:58]This is dividing by 16 * 2, and this is equal to 30 2.

[00:09:05]So, this is the solution here by applying method 2. So, both method 1 and method 2 yields the same value of 81 root 6 over 32.

[00:09:19]Kindly follow the steps.

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