Germany | Can you solve this? | Math Olympiad

Added: 2026-05-12

[00:00:00]6 ^ x + 6 ^ x + 6 ^ x / x 2 ^ x + 2 ^ x this is equal to 36. So what is the value of x? Now let's provide a solution from here.

[00:00:16]Now as you can see from the numerator here we have that 6 ^ of x is common.

[00:00:22]Let's factor out 6 ^ x so that we have 6 ^ x / 6 ^ x. This is 1 + 1 + 1 cross the parenthesis. This is divided by 2 ^ x which is common. Let's factor out 2 ^ x into the parenthesis. This is 1 + 1 and this is equal to 36.

[00:00:50]So this is 6 ^ x * 1 + 1 + 1 this is 3 / 2 ^ of x * 2 this is equal to 36.

[00:01:07]The next step is to multiply on both sides by 2 over 3 by 2 / 3. So that now we eliminate two here and also we eliminate three. So that we have 6 ^ of x / 2 to the^ of x this is equal to 36 / 3 this is 12 12 * 2 this is = 24.

[00:01:35]The next step is that 6 ^ x / 2 ^ x.

[00:01:39]This is in the form of a ^ n / b ^ n which we can express as a / b raised to ^ of n.

[00:01:53]Applying this property then here we have 6 / 2 ra to the^ of x. This is equal to 24.

[00:02:05]Let's simplify here. 6 / 2 this is 3. So we have 3 ^ x this is equal to 24.

[00:02:13]Now to solve for the value of x let's introduce logarithm on both sides. We have log 3 raised to the^ of x. This is equal to log 24.

[00:02:27]Now we have that 3 to the^ x. This is in the form of log a raised to the power of b which we can express as b rogue rog a.

[00:02:39]Applying this power property this for start log 3 to the^ of x becomes x rog 3.

[00:02:47]This is equal to rogue 24.

[00:02:52]Now let's divide on both sides by rogue 3. Here we have rogue 3.

[00:03:01]Now let's simplify rogue 3 and rogue 3 here. So that now x is equal to rogue 24 / rogue 3.

[00:03:14]The next step is that we can express 24.

[00:03:18]This is the same thing as 8 * 3 which you can also express as 2 ^ of 3 multiplying by 3. So x is equal to rogue 2 ^ of 3 * 3 / 3.

[00:03:42]The next step is that 2 ^ 3 * 3. This is in the form of log a * b which we can express as log a + log b.

[00:03:57]Applying this logarithm property we have x is = 2 ^ 3 / ro 3 then plus rogue 3 / rogue 3.

[00:04:15]So let's simplify 3 and 3 here. So this is one. So that we have x is equal to this is 1 + now let's apply the power property here log 2 ^ 3 becomes 3 rogue 2 / rogue 3 we have that log 2 / rog 3 this is of the form rogue a over rogue b this is equal to rogue A to base B.

[00:04:53]Applying this property, this means we have the value of X which is = 1 + 3 2 base 3. So this is the value of X. This is the value of X.

[00:05:12]The next step is to verify if this value of X here satisfies the equation.

[00:05:20]Now let's verify that this value of x here satisfies the equation. Now if you recall we have that 6 raised ^ x + 6 ^ x + 6 ^ x / 2 ^ x + 2 ^ x. This is supposed to give us value of 36.

[00:05:48]So we have that in the numerator here 6 ^ x is common. Let's factor out 6 ^ x.

[00:05:55]So that now here we have 1 + 1 + 1 / 2 ^ x which is common here. Let's factor out 2 ^ x so that we have 1 + 1.

[00:06:11]This should give us a value of 36.

[00:06:16]So this implies that we have 6 ^ of x multiplying by 1 + 1 this is 3 1 + 1 + 1 this is 3 + 2 ^ of x multiplying by 2.

[00:06:29]This is supposed to give us a value of 30 6.

[00:06:36]So we have 6 ^ x / 2 ^ x. This is same thing as 6 / 2 raised ^ of x * 3 / 2. This is supposed to give us a value of 36.

[00:06:52]So this means that we have 3 raised ^ of x * 3 / 2. This should give us a value of 36.

[00:07:05]So let's substitute the value of x.

[00:07:09]We have the value of x as 1 + 3 2 3. Now this is multiplying by 3 / 2.

[00:07:21]This should give us a value of 36.

[00:07:27]Now 3 is a power. So we can express this as 3 ^ 1 + log 2 ^ of 3 to base 3 everything here multip by 3 / 2. This should give us a value of 36.

[00:07:47]Now we have 3 ^ of 1 + 2 ^ 3 to base 3.

[00:07:52]This is in the form of a ^ n + m which we can express as a ^ n * a ^ of m.

[00:08:02]Applying this exponent property then we have 3 ^ 1 * 3 raised ^ 2 ^ of 3 to base 3.

[00:08:13]This is multip 3 / 2 and this should give us a value of 36.

[00:08:22]We have 33 ^ 2 ^ 3 to base 3. This is in the form of a raised to the power of log m to base b m to base a. So this should give us a value of m.

[00:08:38]Applying this logarithm property we have a to the power of log m to base a. This should give us a value of m.

[00:08:48]So this implies that 3 to the^ of 2 ^ 3 to base 3. This should give us a value of 2 ^ of 3 and this is equal to 8. So let's substitute 8. So we have 3 multiplying by 8 multiplying by 3 / 2. This should give us a value of 36.

[00:09:20]So 8 * 3 this is 24 multiplying by 3 / 2 this should give us a value of 36 24 / 2 this is 12 so that 12 * 3 this is equal to 36 which is equal to 36.

[00:09:42]So that we have the left add side is equal to the right add side and this affirms that the value of x in this case which is 1 + 3 2 to base 3 satisfies the equation. So kindly follow the steps like this video and kindly subscribe to this channel. See you in the next video.