Germany | Can you solve? | A Nice Exponential Maths Problems |

本站添加: 2026-05-09

[00:00:00]Hello everyone. Welcome to Rasha's Classroom. Today we are solving a interesting German math Olympiad question which is 2 to the power x + 1 + 2 to the power 3 - x is equal to 17. x is equal to what? How to solve this interesting exponential math problem? So first of all, our math solution, I use Here, I use index law.

[00:00:30]We all know that a to the power n + m which is a to the power n * a to the power m. And we know that other formula, a to the power - n which is 1 over a to the power n. I apply both formula here.

[00:00:51]So this expression if I apply this math formula here, so it will be 2 to the power x * 2 to the power 1 then this is 2 to the power 3 * 2 to the power - x is equal to 17.

[00:01:10]I apply this math formula here.

[00:01:13]So this expression, it will be 2 to the power x * 2 + 2 to the power 3 which is 8 * but 2 to the power negative x, if I apply this math formula, it will be 1 over 2 to the power x is equal to 17.

[00:01:32]Then this is 2 * 2 to the power x, 2 * 2 to the power x then + 8 * 1, this is 8 over 2 to the power x is equal to 17.

[00:01:46]Now, this is 2 to the power x, this is 2 to the power x.

[00:01:49]So at this moment, let let 2 to the power x is equal to u.

[00:01:58]Now, I substitute this value here.

[00:02:05]If I substitute this value here, so this expression it will be 2 u plus 8 over u is equal to 17.

[00:02:17]Now, this is uh u this is u, so I take here is least common value is u. So, this divided this here is one one divided u u times 2 u it will be 2 u squared.

[00:02:30]And this u divide this u it will be one one times eight this is eight is equal to 17.

[00:02:37]Now, I multiply both side by u, so it will be 2 u squared plus eight then here 17 u.

[00:02:48]Now, we are find out a nice quadratic equation uh which is 2 u squared minus 17 u. I just move on this 17 in this side it will be negative 17 plus eight is equal to zero. Now, at this moment I do here is middle factor because of that it is a nice quadratic equation. So, this is 17, but this is eight times two this is 16. So, this is 2 u squared minus 16 u minus u plus eight is equal to zero.

[00:03:22]Now, here is 2 u is common. So, if I take u is common this divide this this is u minus this divide this this is eight minus one is common this is u minus eight plus minus this is minus is equal to zero.

[00:03:37]Now, this is u minus eight this is u minus eight. So, at this moment you can say here easily u minus eight is common then this divide this this is 2 u this divide this this is minus one is equal to zero.

[00:03:50]Then, here u is u - 8 is equal to zero.

[00:03:55]Then, 2 U - 1 is equal to zero. Then, U is equal to 8. And, you can say easily here, U is equal to If I move on this -1 in this side, it will be positive 1.

[00:04:07]Then, if I divide both side by 2, we are find out 1 over 2. So, at this moment, remember that we are substitute here is U is equal to 2 to the power X.

[00:04:18]So, I take this value here because of that our target X is equal to what? So, U is equal to 2 to the power X. But, this 8 it will be 2 to the power 3.

[00:04:27]Now, according to exponential formula, this is two, this is two, base is same, so exponent is equal, so X is equal to three.

[00:04:34]But, this is 2 to the power X. But, this is 1 over 2, this is 2 to the power -1.

[00:04:39]Now, at this moment, you can say here X is equal to -1.

[00:04:44]So, we are find out X is equal to three, X is equal to -1. This is our final answer in this tricky exponential maths problem. Let's verify. We are find out X is equal to three and X is equal to -1.

[00:05:00]But, our question it will be 2 to the power X + 1 + 2 to the power 3 - X is equal to 17.

[00:05:11]Then, I take X is equal to three. So, if I take X is equal to three, so 2 to the power 3 + 1 + 2 to the power 3 - 1. Sorry, X is equal to three.

[00:05:25]Then, is equal to 17.

[00:05:28]Now, at this moment, this is 2 to the power 4 and this is 2 to the power 0 is equal to 2 to the power 4, this is 16 and 2 to the power 0, this is 1. 16 + 1, this is 17. So, left hand side and right hand side both side is equal.

[00:05:44]Now, if I take -1, okay? Now, at this one I take here is -1. Let's verify.

[00:05:50]This is 2 to the power -1 +1. Because of that x is equal to -1.

[00:05:57]I take this one here + 2 to the power 3 -x. But x is equal to -1. So I take -1 here is equal to 17.

[00:06:07]So at this moment our left hand side which is 2 to the power 0 + 2 to the power minus minus this is plus I can say easily this is 3 + 1. This is 4 and is equal to 17.

[00:06:21]Now this is 16. 16 + 1 because of that a to the power 0 it will be 1. 1 + 16 this is 17 is equal to 17. So left hand side and right hand side both side is equal.

[00:06:32]So our answer is x is equal to 3, x is equal to -1. If you enjoy this math problem please subscribe my YouTube channel. Thank you everyone.

[00:06:40]Have a good day. Take care everyone.