A Trending Nice Math Olympiad exponential question| how to solve for x.

My channel, um, let's solve this on the board. We are asked to solve for x in this particular equation. Now this is an exponential equation, so we are going to solve it using the laws of indices. Okay, first of all, the square root of x is the same as x raised to the power of one half. Okay, the square root of x or the square root of anything is the same as that thing raised to the power of one half. that means that we can solve this as what 100 raised to the power of root x equals 1000 raised to the power of x raised to the power of one half is the same as 100 raised to the power of root x equals 1000. This multiplied by this will give us x over 2. Now, let's put them in the same base. Let's put them in the same base. we'll have what 10 to the power of 2 root x is equal to 10 to the power of 3 raised to the power of x over 2, so we can rewrite this as what, where you have something like this for example, a to the power of n raised to the power of m is the same as a to the power of n times m, so let's apply that here. This can be written as 10 to the power of 2 times root x is equal to 10 to the power of 3 times x over 2 is 3 x over 2, okay, they have the same base and once they have the same base, you can equate the exponents. Okay, so we can now write that 2 times the square root of x. is equal to 3x over 2. All right, now let's cross multiply: 2 times 2 root x gives us 4 root x. is equal to 3 times x, which gives us 3x. All right, now let's square both sides of this equation. Let's square both sides of this equation so that we can eliminate this radical sign. So it's going to be 4 root x squared equals 3 x squared. Now 4 squared will give us what, 16? root x squared gives us x, 3 squared gives us 9, x squared is 9 x squared, okay, that means we can write this as what? 16x minus 9x squared is equal to 0. we can factor out x, so write x times (16 minus 9x) equals 0; so this means that We can write that x is equal to 0 or 16 minus 9x is equal to 0. So let's solve the second one; this one is already solved, so let's solve this one to get the value. of x if this is the first time you're coming across this channel, please don't forget to Click on that subscription button and also don't forget to ring your notification bell—don't forget to comment either. Like and share, okay, and stay tuned because we are going to prove the solution that we got. So if we should solve this, let's take this to the other side of the equation; we'll have that 16. is equal to 9x; if we divide both sides of this equation by 9, we will have that x is equal to 16 over 9. Now let's prove our solution. Let's prove our solution. The first one says that x equals 0, right? So that means we can write that 100 raised to the power of the square root of 0. is equal to the square root of 1000 raised to the power of 0, okay, this is the same thing as 100. The square root of 0 is 0, and the square root of 1000 to the power of 0 is 1, yes, 1, any number Any number to the power of 0 is 1. What is 0 to the power of 0? Tell me in the comment section because There's so much controversy surrounding that one; what is 0 to the power of 0, anyway? All right now. Since we said any number to the power of 0 is 1, that means this side is 1, which is equal to square. The square root of 1 is also 1, so the left-hand side is equal to the right-hand side, which means definitely 0 is a solution; now let's prove the second one. The second one states that our x is equal to 16 over 9. This is where we need to do some mathematics to prove that yes, x is 16 over 9, so let's do that so we have 100 times the square root of 16 over 9 equals the square root of 1000 raised to the power 16 over 9, all right, so this can be written as 100. The square root of 16 is 4, and the square root of 9 is 3. Is 3 equal to this same thing as 1000 raised to the power of 16 over 9, everything raised to the power of half because of this square root sign, so 2 we divide itself to give you 1, and we divide 16 to give you 8, all right, so we have 104 over 3 is equal to 1008 over 9; they are not equal, right? No, they are not. So let's keep proving this; this is the same thing as 10 to the power of 2, right? Raising to the power of 4 over 3 equals 10dü 1000 is the same as 10 to the power of 3. raised to the power of 8 over 9; now this can be written as the left-hand side of the equation can be written as 10 to the power of what 2 times 4 is 8 over 3 is equal to 10 then 3 we divide this to give you 3, which you also give you 8 over 3, so the left-hand side is equal to the right-hand side. Thank you so much for watching, and still have a lovely day. Bye.