How to Find the Inverse of a Binary Operations

Added: 2026-05-26

[00:00:01]In order to find the inverse of this element, what we are simply going to do is to take that operation X and then we say on the operation. Now, in place of this Y, we are going to put the inverse. Let the inverse of the element be X raised to power minus one.

[00:00:21]That is the inverse of X. Let it be X raised to power minus one.

[00:00:25]And that will be equal to. So, we are going to use this expression because this is expressed in this form. So, anywhere we see Y, we are going to put the inverse, which is X raised to power minus one. So, this is X. I'm going to write it.

[00:00:39]And then plus, in place of Y, just as we have replaced Y with the inverse of the element. So, in place of Y, >> [snorts] >> we are going to put X to the power minus one. So, I have X to the power minus one.

[00:00:51]And then minus three, which is there.

[00:00:55]And then X.

[00:00:57]And then in place of this Y, as usual, I'm going to put X to the power minus one. That is the inverse. And everything should be equal to the identity element, which we are told is zero. So, everything should be equal to zero, which is the identity element. So, this is how to carry out the inverse operation. And then now, you can leave this first part and simply equate these two and solve for the inverse.

[00:01:26]So, I'm going to write X plus X to the power minus one minus three X and then X to the power minus one is equal to zero. That is equating these two parts now in order to solve it. So, I can simply leave these two terms that are like terms. They have X raised to power minus one. So, I can simply leave them here. Minus one minus three X raised and multiplied by X to the power minus one is equal to this x now can come to this side becomes minus x.

[00:02:03]And then I can factorize this now. X raised to power minus one is common, so I put x raised to power minus one bracket open. This divided by this I have one minus this divided by this I'm left with three x.

[00:02:18]And is equal to minus x. So in order to get the inverse now I'm simply going to divide both side by one minus three x and then divide both side by one minus three x. So that this will cancel this and I'm left with the inverse which is what I am looking for and that will be equal to negative x which is minus x all over one minus three x. And this is the final answer. Hope you enjoyed this video.

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