Harvard University Entrance Exam Question | Can you solve ?

Added: 2026-05-20

[00:00:01]Hello, welcome back once again. Today we're going to solve this interesting exponential equation 16 to the power of X plus 4 to the power of X is equal to 36 to the power of X.

[00:00:15]So let's get started.

[00:00:17]Now for us to start we're going to divide each term from both sides by 4 to the power of X. So here we obtain 16 raised to the power of X divided by 4 raised to the power of X then plus 4 raised to the power of X divided by 4 raised to the power of X.

[00:00:36]So this is equal to 36 raised to the power of X divided by 4 raised to the power of X. Now using this property we know A raised to the power of M divided by B raised to the power of M is equal to A divided by B whole raised to the power of M.

[00:00:53]So therefore this one here becomes 16 by 4 whole raised to the power of X plus here this one is one. So this is equal to 36 by 4 whole raised to the power of X.

[00:01:09]Now here this simplifies nicely. So from the right hand side here we just have 9 then raised to the power of X. So this is equal to from the left hand side we just have 4 raised to the power of X plus 1.

[00:01:21]Now subtract 4 to the power of X from both sides to get 9 to the power of X minus 4 to the power of X is equal to 1.

[00:01:30]Now we know this base here is 3 squared.

[00:01:33]So with this X here becomes 3 to the power of 2 X, right? So this is 3 to the 2 X minus here this is 2 squared. With the power here that becomes 2 to the 2 X. So this is equal to 1. Now let's make the following substitution. So we're going to let 2 X equals Y.

[00:01:55]So that our X value will be y divided by 2.

[00:02:00]So from here we get 3 raised to the power of y minus 2 raised to the power of y is equal to 1. And we can see one possibility here, which is obvious, right? And what's that? That y can only be equal to 1 when we're talking about positive integer, right? Because 3 raised to the power of 1 is 3 minus 2 raised to the power of 1 is 2. So 3 minus 2 that will give us 1. So therefore y is equal to 1.

[00:02:32]Now let's go back remember that x is equal to y divided by 2, right? So from here this implies from here that x is equal to y divided by 2. So this is giving us our solution here that x is equal to 1 divided by 2, which is half or 0.5.

[00:02:53]So this right over here is our solution and this indeed satisfies the given equation. Thank you for watching. If you enjoyed the video, please kindly subscribe to this channel with a like, comment, and share.