Secret trick to solve radical equation | Olympiad mathematics |

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[00:00:00]solution to this radical exponential equation.

[00:00:03]The equation says the square root of 2 to the power of x - 3 plus the square root of 2 to the power of x - 5 equals 6. So, we are asked to find the value of x.

[00:00:14]This is what to do.

[00:00:16]Now, we know that this can be written as 2 to the power of x - 3 all to the power of 1 over 2. We are transforming the square root to the power of 1 over 2.

[00:00:26]plus 2 to the power of x - 5 all to the power of 1 over 2. All this is equal to 6.

[00:00:35]Now, let's multiply x - 3 by 1 over 2.

[00:00:38]So, this becomes 2 to the power of x - 3 over 2 plus 2 to the power of x - 5 over 2 is equal to 6.

[00:00:51]Now, look at the way I want to express this um x - 3.

[00:00:56]I can rewrite this - 3 as - 5 + 2, which become 2 to the power of x - 5 plus 2 over 2 because - 5 + 2 is - 3 plus 2 to the power of x - 5 over 2 equals what? 6.

[00:01:14]>> [snorts] >> Now, let me write this as 2 to the power of x - 5 over 2 plus 2 over 2, right? plus 2 to the power of x - 5 over 2 Okay.

[00:01:31]All this is equal to 6.

[00:01:33]>> [snorts] >> Now, 2 over 2 is 1.

[00:01:37]So, that means let me write this as 1 because it is all the same.

[00:01:42]I can also rewrite this from the law of indices. The same base we we add we we multiply.

[00:01:47]When we are multiplying the same base, we add the powers. So, this becomes 2 power of x - 5 over 2 multiplied by 2 power of 1 plus 2 ^ x - 5 over 2 equals what? 6. Because the same base we shall add the powers which results to this.

[00:02:07]Now, what do you observe?

[00:02:09]Um 2 ^ x - 5 over 2 is a common factor.

[00:02:13]So, we shall factorize this equation by using the common factor which is 2 ^ x - 5 over 2. Which means if the whole of this thing divide this, I shall have 2 ^ 1 which is 2 + this divide this, I shall have 1. All of this is equal to 6, okay?

[00:02:31]Now, 2 + 1 is 3. So, this becomes 2 ^ x - 5 over 2 multiply by 2 or by 3, sorry. All of this equals 6.

[00:02:45]Now, divide both sides by 3 and divide here by 3.

[00:02:49]This cancels out. We shall have 2 ^ x - 5 over 2 equals 2. Of course, 2 is the same as 2 ^ 1. The same base we shall equal the powers to become x - 5 over 2 to be equal to 1. Call this over 1 so that you can cross multiply.

[00:03:08]So, this implies that um x - 5 will be equal to 2 * 1 is 2.

[00:03:16]Now, we shall collect things which become our x to be equal to 2 + 5. Because when - 5 crosses this become what? + 5. This is equal to or this implies that our value of x will be equal to 2 + 5 is 7. So, the value of x equals 7. Thank you. Share, follow, and subscribe for more math tips like