Algebra Mastery: Solving Mixed Exponential and Power Equations

Added: 2026-05-17

[00:00:00]Do you believe that most students cannot solve this Olympian mathematics question on the board?

[00:00:04]The question says 9 to the power of x minus x to the power of 4 is equal to 65.

[00:00:11]Let me show you to how to solve this. By the end of this video, you will know the solution to this equation and how to solve this. Okay?

[00:00:20]All right.

[00:00:21]This question demands some logical thinking.

[00:00:24]Okay? You will love this.

[00:00:26]So, what I'm going to do is that look at this 9. This 9 can be expressed as 3 to the power of 2 all to the power of what? x.

[00:00:37]Okay?

[00:00:38]Minus this 4 can be expressed as 2 * 2, which become x to the power of 2 all to the power of 2. This is equal to what? 65.

[00:00:50]Okay? Now, permit me to interchange this power from the product law of indices because if you have a to the power of c all to the power of b, this implies this can also be written as a to the power of b all to the power of c.

[00:01:08]So, from here, this can be also be expressed as 3 to the power of x all to the power of 2 minus x squared all to the power of 2 is equal to 65.

[00:01:22]Now, look at this.

[00:01:24]Observe this is written in a difference of two squares because if you have a squared minus b squared this can be written as a plus b multiplied by a minus b.

[00:01:41]So, this this is written in this form, which can be transformed to this form.

[00:01:46]Okay? That means this can also be expressed as what? 3 to the power of x minus Okay? Plus um x squared multiplied by 3 to the power of x minus x squared.

[00:02:02]Okay, all this is equal to Now, this 65 can be expressed as 13 times 5.

[00:02:11]Okay? Now, what we are going to do is that because this is this is plus and this is minus. So, let let us equate the one with plus to the bigger number and the one with minus to the smaller number because this value must be bigger than this value.

[00:02:25]Okay, if you plug in the value of x to this equation, this must be bigger than this. So, let's equate this to the bigger number and this to the smaller number. So, this equation becomes um 3 to the power of x plus x squared is equal to 13. You can call this equation 1.

[00:02:41]Okay? Then, we shall have 3 to the power of x minus x squared to be equal to 5.

[00:02:47]You can call this equation 2.

[00:02:50]Now, we are going to solve this simultaneously though it involve one variable.

[00:02:55]To solve this, you are going to add equation 1 to equation 2.

[00:02:58]So, adding equation 1 to equation 2, okay?

[00:03:01]If you add this to this, we shall have two times of uh 3 to the power of x. Is that right?

[00:03:12]Now, if you add this to this, that means this has gone.

[00:03:15]So, this is equal to 13 plus 5 is what?

[00:03:19]Is 18.

[00:03:21]Right? So, divide both sides by two to clear the two in the left hand side. So, we shall have 3 to the power of x is equal to 18 divided by two is nine.

[00:03:31]So, what next? Nine can be expressed in the base of three, so we shall have 3 to the power of x is equal to 3 squared. So, x is equal to two and that happens to be the final answer. Thank you for watching. Share this video and also subscribe to our channel for more math tips like this. Thank you.