Math Olympiad Training: Solving Tower Exponents

Added: 2026-05-10

[00:00:00]Do you believe that most people cannot solve this exponential equation?

[00:00:04]By the end of this video, you will know the solution to this equation.

[00:00:08]All right. We have 2 to the power of 8 to the power of root of 3 all to the power of x is equal to 4.

[00:00:15]Now, first to solve this, look at 4 and 2.

[00:00:20]4 can be expressed in the base of 2.

[00:00:23]Right? So, we shall have 2 to the power of 8 to the power of root of 3 all to the power of x is equal to 2 to the power of 2.

[00:00:37]Now, apply this law. a to the power of b is equal to a to the power of c, which implies that b is equal to c.

[00:00:45]So, this equation become 8 to the power of the root of 3 all to the power of x is equal to 2.

[00:00:56]Okay?

[00:00:58]Now, this has a natural power of 1.

[00:01:02]Now, some people will make a mistake.

[00:01:04]Watch this out.

[00:01:06]Don't ever try to make this error while solving this. Because some people will notice that will will will like will think that this x is another power of this root of 3, not knowing that this is multiplying each other.

[00:01:19]Okay? Just look at this.

[00:01:21]So, we can choose to interchange this power because from the protocol of indices, but anyways, I'm not going to do that because of this 8.

[00:01:29]Now, we are we are going to express this 8 to be in this form, which become 2 to the power of 3 is 8 multiplied by the root of 3.

[00:01:40]Okay? Now, this will multiply this. We shall have um That's what we shall have.

[00:01:49]Okay?

[00:01:51]This is equal to 2 to the power of 1.

[00:01:54]From this law also, this become 3 root of 3 x is equal to 1.

[00:02:05]Now, this is what to do.

[00:02:08]Do you believe that I can divide both sides by 3 root of 3?

[00:02:11]So, if I divide this side by 3 root of 3 also divide this side by 3 root of 3. This will neutralize this.

[00:02:23]We shall have x to be equal to 1 over 3 root of 3.

[00:02:29]And that happens to be the final answer.

[00:02:32]You can choose to multiply both sides or to multiply this thing by root of 3 and find the I multiply both sides by 3 root of 3.

[00:02:41]No, not both sides. Multiply this by like 1 over 3 root of 3 times root times root of 3. Multiply here by what?

[00:02:53]root of 3.

[00:02:55]So, that we shall have um this equal to root of 3.

[00:03:00]Okay?

[00:03:01]If you multiply these together, this become 3 multiplied by root of 3 times root of what? 3.

[00:03:10]Okay? So, we shall have this is equal to root of 3 divided by 3.

[00:03:15]If you multiply these together, we shall have uh root of 3 all squared.

[00:03:21]This will cancel out this. This is equal to the root of 3 divided by what?

[00:03:27]3 * 3.

[00:03:29]Okay? So, 3 * 3 is root of 3 divided by what? By 9. So, this happens to be the final answer. Either this or this. Both answers are still correct.

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[00:03:45]Thank you.