Nice Algebra Problem | Exponential Equation | Math Olympiad

Added: 2026-05-16

[00:00:00]If you don't watch this video to the end, you have yourself to blame during your exams. Watch this.

[00:00:07]You are asked to find the value of x given this kind of exponential equation involving radical.

[00:00:12]We have 5 to the power of the root of x plus 5 to the power of the root of x is equal to 8 to the power of root of x.

[00:00:19]Can you solve this?

[00:00:20]Anyways, let me help you out in case you come across this.

[00:00:24]It is very simple to comprehend. You just need some beautiful steps to apply and get your final answer.

[00:00:29]Now, I am going to factorize the left-hand side because we have 5 to the power of the root of x plus 5 to the power of root of x. So, the common factor is one of these.

[00:00:40]Let's see. So, we shall have 5 to the power of the root of x as a common factor, which means if this divide this without remainder, the answer is one. You put a plus sign.

[00:00:50]This divide this without remainder, that means we have one.

[00:00:53]All this is equal to 80 to the power of the root of x.

[00:00:58]1 + 1 is 2.

[00:01:00]So, we shall have 5 to the power of the root of x multiplied by 2 to be equal to 80 to the power of what? The root of what? x.

[00:01:11]I don't want to divide both sides by 2, but I want to divide both sides by 5 to the power of the root of x so that we can clear the the the unknown to one side cuz it is important to clear the unknown to one side to make sure that you you find the unknown.

[00:01:27]So, now, we shall divide both sides by 5 to the power of the root of x, which becomes divide here by what? 5 to the power of the root of x and divide here by 5 to the power of the root of x.

[00:01:40]So, if you divide both sides by 5 to the power of root of x, observe that if this cancels out this, you are left with 2.

[00:01:46]That means 2 will be equal to 80 to the power of the root of x divided by what? By 5 to the power of the root of x.

[00:01:56]Let's apply this law of indices which state that if you have um a to the power of b divided by c to the power of b, this implies that a over c all to the power of p. So, we are going to transform this using this. So, this become two is the same thing as saying two to the power of one to be equal to 80 divided by five all to the power of the root of x because x is common to both numerator and denominator.

[00:02:29]So, 80 divided by five is 16, which means two to the power of one is equal to 16 to the power of the root of x.

[00:02:40]Now, watch this.

[00:02:41]I can write 16 as two to the power of four, which means we shall have two to the power of one to be equal to two to the power of four is 16 multiplied by the root of x.

[00:02:53]Applying this law of indices, if you have a to the power of b to be equal to a to the power of c, that means b is equal to c because the base are expressed in the same form.

[00:03:04]So, given that this is equal to this, that means we shall have one to be equal to four root of x. I am looking for a way to clear the radical in the right hand side. How can I achieve that? You can achieve that by multiplying both powers all to raise both powers um to the power of two.

[00:03:23]Like you multiply both powers by two, I multiply here by two. Meanwhile, one to the power of two means one times one, which is equal to one, to be equal to Now, this square this square above affect both four and root of x, which means we shall have four squared multiplied by the root of x squared when we remove the bracket, which means this neutralize this. That means the square cancel out the square root, we are left with x, and 4 squared is 16, which means um 1 will be equal to 4 squared is 16 multiplied with x we shall have 16 x. To find the value of x divide here by 16 and divide here by by 16. This cancels out the value of x will be equal to 1 over 16. And that happens to be our final answer. Please share this video, follow us and subscribe to our YouTube channel for more math tips like this.

[00:04:18]Thank you.