Olympiad Mathematics | Indian | can you solve this?

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[00:00:00]Okay, if you're ready, let's solve this problem here.

[00:00:04]This is some a quick one.

[00:00:09]We have X over two all to the power of three equals 64.

[00:00:18]So, how do we solve this problem here?

[00:00:21]You know, we can always split what we have here. You see that the base you know, the power will be for both numerator and denominator.

[00:00:30]So, this is the same as X to the power of three over two to the power of three equals 64.

[00:00:39]Okay, since we are writing everything in the power of three let's write 64 in the power of three because 64 is a perfect cube.

[00:00:50]Write X to the power of three over two to the power of three equals 64 is four to the power of three.

[00:00:58]4 * 4 is 16, 16 * 4 and it's 64.

[00:01:03]Right?

[00:01:04]Now, cross multiply and we have X to power three to be equal to two to power three times four to power three.

[00:01:15]And at this point, we shall apply a law of indices that says A to the power of M times B to the power of M is the same as A times B both to the power of M.

[00:01:31]Right? So, if this is true, then we shall work on the right-hand side to get 2 * 4, that is eight. Then, it is to the same power of three.

[00:01:46]Simple, right?

[00:01:48]So, at this point, we have from two cubes we have X cubed equals eight cubed. So, ordinarily, you can agree with me that our x is equal to eight.

[00:01:59]But, that will give us only one solution.

[00:02:03]>> [snorts] >> So, we have to do this, bring this to the left, we have eight to the power of three, which is equal to zero at this point.

[00:02:12]And then, we are going to understand that a to the power of three minus b to the power of three is an identity, which is equal to a minus b and we'll multiply a squared plus ab plus b squared.

[00:02:34]Right? So, this is the identity for difference of two cubes.

[00:02:38]Now, our a minus b shall be x minus eight.

[00:02:44]Then, a squared shall be x squared.

[00:02:47]ab is going to be um x times eight, and that is eight x.

[00:02:54]Then, b squared is going to be eight squared.

[00:02:58]Yes, b squared is going to be um eight squared.

[00:03:04]And let me write it as eight squared.

[00:03:08]If you want, you can write it as 64, but I prefer writing it as eight squared.

[00:03:13]So, at this point, we apply our zero product rule.

[00:03:20]A lot of students don't know when to apply this.

[00:03:24]We apply zero product rule when you have two terms to multiply to get zero.

[00:03:31]x minus one is considered a term.

[00:03:34]Then, x squared plus eight x plus um eight squared is also a term. So, we're going to say that either this is equal to zero or this is equal to zero.

[00:03:46]So, let's go.

[00:03:50]Okay, so from here, applying the zero product rule, x - 8 is equal to zero. I'll pick this after.

[00:03:59]And from here, our x is going to be zero plus eight. Meaning that the value of x is eight.

[00:04:07]This is our first solution.

[00:04:09]And we need to get two more solutions, which will come out of this.

[00:04:15]We have x squared plus eight x plus eight squared equals zero.

[00:04:23]I told you, if you want, you can write your you can write your eight squared as 64.

[00:04:30]It doesn't change a thing.

[00:04:33]Now, we are going to solve this quadratically using quadratic formula.

[00:04:38]And that formula has a, which is the coefficient of x squared, and it's one.

[00:04:44]It has b, which is the coefficient of x, and it's eight.

[00:04:49]And then it has c, which is the constant, and in this case, it is eight squared.

[00:04:56]So, what then is the formula?

[00:04:58]The formula is x equals minus b plus or minus the square root of b squared minus four ac.

[00:05:10]All of this is divided by two times a.

[00:05:14]Okay, so once you're able to remember this formula, the work is almost done.

[00:05:19]Just direct substitution.

[00:05:21]Our x will be minus b, which will now be minus eight.

[00:05:26]Right? Plus or minus, sorry, I'm writing out of sight. So, our b squared is eight squared now.

[00:05:34]Minus four times one, cuz a is one, and c is what?

[00:05:40]C is eight squared.

[00:05:43]So, we divide all of this by two times one, which will be which will still be two.

[00:05:50]Now, going on from here, we have x to be minus eight plus minus the square root of If you look at this term and this term, you know, four times one times eight squared is considered a term.

[00:06:06]And what is common to them is eight squared. So, we write eight squared as the common factor.

[00:06:13]Eight [snorts] squared divided by itself is one. Then, this the whole of this is going to be four times one, which is four.

[00:06:21]Because eight squared here is already out.

[00:06:24]We divide by two.

[00:06:27]Now, to go on, we shall have x to be equal to minus eight plus minus We have the square root of eight squared times minus three.

[00:06:42]Yes.

[00:06:43]And we're dividing all of this by two.

[00:06:46]And I hope you remember this um rule that if you have the square root of a times b, you can split it up and get the square root of a multiplied by the square root of b.

[00:07:00]Yes, I said it. We can always do this.

[00:07:03]So, from here now, guess what we are going to do.

[00:07:07]We will now work on this like this.

[00:07:09]So, permit me to remove this.

[00:07:12]So, our x will be minus eight plus or minus we have the square root of eight squared multiplied by the square root of minus three.

[00:07:25]And all of this is still over two.

[00:07:28]Okay, so we can find the square root of this separately. Let's do it.

[00:07:34]Okay, so from here now, we shall have x to be equal to minus eight plus or minus um this can take this out. so we have eight.

[00:07:44]Then we have square root of three multiplied by square root of -1.

[00:07:51]So, what I've done is to split this.

[00:07:53]I will divide all of these by two.

[00:07:56]Okay, I want to show all the steps.

[00:07:59]Our X now is minus eight plus or minus where we already have eight, but square root of -1 is I.

[00:08:07]So, eight times I will give us eight I.

[00:08:10]Then we have our root three divide this by two.

[00:08:14]So, that at the end of the day two here can divide the two numerators. So, that X will be minus eight divided by two, that is minus four.

[00:08:25]Plus or minus eight I divided by two is four I.

[00:08:29]Then this is root three.

[00:08:32]Now we have a two-in-one solution.

[00:08:37]Let's bring the three solutions together.

[00:08:40]We got X to be equal to eight as our first solution.

[00:08:45]And from this last part we have um we have minus four plus four I root three as our second solution.

[00:09:00]Then the third solution still comes from the last part.

[00:09:05]Where we have minus four minus four I.

[00:09:08]Then we have root three. So, these three are the solutions to the equation.

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