AMERICAN || Can You Solve this Entry Olympiads IN 2 SECS?

Added: 2026-05-25

[00:00:00]Hello, you're welcome to solving this exponential problem. X to the^ x - 2 = 16. Solving for the value of x. So we have x^ x - 2 = 16 is the same as having 4 to the^ of 2.

[00:00:20]So what we are going to do will be to express this particular 4^ 2 as this. So we have x ^ x - 2 equ= 4 to the^ of 2 / 1.

[00:00:43]Then we have * 4 ^ 2 / 4 to the^ 2.

[00:00:57]So by expressing it this way remember this can easily eliminate this and we have 4 to the^ of two right here. So we haven't really changed anything.

[00:01:12]Now from here we can multiply the numerator.

[00:01:16]So what we have is x ^ x - 2 = 4 ^ 2 * 4 the^ 2. Remember if you have a power b * a + c equals a power b + c. So 4 to the^ of 2 * 4 the^ of 2 we are going to have 4 to the^ of 2 + 2 / 1 * 4 to the^ of 2 we have 4 to the^ of 2.

[00:01:53]So from here x^ x - 2 equals this is 4 ^ 2 + 2 that is 4 / 4 to the^ 2.

[00:02:10]Now consider this as well. If you have a to the^ b / a^ c this equals a the^ b minus c.

[00:02:21]So applying that right here as well you have x to the^ x - 2 equals this. Now but before then before then I can just apply this identity right here. So this means I'm going to have x ^ x / x to the^ of 2 because look at what happens here and this equals 4 to the^ 4 / 4 the^ 2.

[00:02:57]So we apply this identity right here and we have x ^ x / x ^ 2.

[00:03:05]Now look at what happens here.

[00:03:10]This is x to the^ of x and this is 4 to the^ 4 and also this is x to the^ of 2 and this is 4 to the^ of 2.

[00:03:23]So by comparison if you have a 3 a = 3 that means a = 3. So x to the^ of x = 4 to the^ of 4 and x^ of 2 = 4 to the^ of 2 that means x = 4.

[00:03:44]So this becomes our solution for x.

[00:03:49]We can decide to verify.

[00:03:56]we have x = 4 and this is x to the^ of x - 2 = 16.

[00:04:05]Let's substitute four for x. So we have 4 to the^ of 4 - 2 this = 16.

[00:04:17]4 - 2 is 2. So we have 4 to the^ of 2 = 16. And 4 to the^ of 2 is 4 * 4.

[00:04:29]This = 16.

[00:04:32]And you multiply 4 * 4 that is 16. And this equals 16.

[00:04:41]So right now as long as the left hand side of this equals the right hand side of this.

[00:04:48]So this particular value here satisfies the equation. And that is our solution.

[00:04:57]So the solution to this is x = 4.

[00:05:04]So that is it. We have x = 4 as our answer.

[00:05:09]So that's it. Thank you and goodbye.