Harvard University interview question| Can you pass this exam?

Let's solve for the value of n. You're welcome. First, let's square both sides of this equation.重构 because my intention is to eliminate this square root symbol, okay, and the only way to achieve that is to square both sides of this equation so I will have nine raised to the power of root n squared is equal to the square root of 27 to the power of n squared, okay? Now what we have here now is nine. multiplied by okay sorry because we have what two powers on a number when you have a to the The power of m raised to the power of n is equal to a to the power of m times n. Alright. So we are going to relate this now to what we have here. So this is something as nine to the is equal to this is going to cancel this; we have 27 to the power of n. All right, now when you look At this point, we can write 9 as 3 to the power of 2, right? Sorry, 3 to the power of 2, now raised to the square root of 2 to the power of n is equal to 3 to the power of 3 to the power of n I would like you to Stay put and see how I prove the answers we got, okay? Because to prove that the answers are Correct, this is another math problem in itself, so don't miss the fun, okay? All right, now we can write this as what 3 to the power of when you have two powers you can multiply them together so this is 2 times to give us 4 root n is equal to 3 to the power of 3n, okay? All right, now we have the same base. We can equate the powers, therefore we can write that the fourth root of n is equal to 3n. Now we can square both sides. both sides of the equation because of this root here, okay, so if we square, we have 4 root n squared is equal to 3n squared, okay, so what we have here now is 4 squared multiplied by root n. squared is equal to 3 squared multiplied by n squared, okay? Now 4 squared gives us 16. Multiplying by this, we see that what remains is n equal to 3 squared, which gives us 9. and we have n squared, okay, so this will give us 16n is equal to 9n squared. Now let us bring Bring them together; let us bring them together. If we bring them together, we have 16. This will cross. to become minus 9n squared is equal to 0, okay. Now, what do they have in common? What do they have? In common is what n, so when 16n is divided by n, we are going to have 16 minus when 9n squared. divided by n, we have 9n equals 0. As I said, the main point here is not even to get the For the values of n, the main thing is to prove that they are correct, but before that, please click on the Subscribe to the channel and turn on the notification bell if this is your first time visiting like on this video and on this channel, please. Thank you so much, and don't forget to give us a like. Oh, if you're actually enjoying the video, whether it's the first or the second or Alright, let's continue. Now we can say that n is equal to 0, right? And that 16 minus 9n is equal to 0, so this is the first value of n. Now from here we can say that 16 is equal to minus 9n will cross over to give us plus 9n. All right, now to get the value of n, we are going to divide both sides of the equation by 9, so we are going to have 16 divided by 9 is equal to 9n divided by 9; now we can see this, and we have that the second value of n is equal to 16 over 9 now to the most important and the most interesting aspect of this video Let's check our values. The equation says 9 raised to the power of root n is equal to square. the square root of 27 raised to the power of n, now for n equals 0, because we got the first value of n to be 0, right? So for n equals 0, you can see that we can write 9 raised to the power of root 0. is equal to the square root of 27 raised to the power of 0; right now we have 9, the square root of 0 is 0. So we have 0 here is equal to the square root of any number n to the power of 0 is equal to 1, okay. any number to the power of 0 is 1, so 27 to the power of 0 is 1. Now, 9 to the power of 0 since n to the power of 0 is equal to 1; therefore, 9 to the power of 0 is also equal to 1. square root of 1 now 1 is equal to 1 the left hand side is equal to the right hand side so you I can see that n equals 0 satisfies the equation. Now let us check the second example. The second solution that we obtained, okay, we also have, excuse♌ me, okay, now for n equals which It's the main one, 16 over 9. Let's prove that n is equal to 16 over 9. Okay, this one is even too. A tricky one, not difficult in itself, but tactical—if you're not careful, you won't get it. so the equation says that 9 raised to the power of root n is equal to square root of 27 raised to the power of n. Okay, now can we prove that? So that means that 9 to the power of 16 over 9 square root Okay is equal to the square root of 27 to the power of 16 over 9. Right now, this same thing as 9 to the power of the square root of 16 divided by the square root of 9 is equal to the square root of 27 to the power of 16 over 9, okay, all right. So we have 9 to the power of 16. The square root of 16 is 4. and the square root of 9 is 3 is equal to now the square root of x, that is the square root of any number, square The square root of x is equal to x to the power of one half, okay? x to the power of one half—this is a power, all right? so we can write this as 27 to the power of 16 over 9, all raised to the power of one half Now we have 9 to the power of 4 over 3 equal to 27 squared. We divide this to get 8, so 27. Now let's put them in the same base, yes. So when we do that, we have 3 divided by 3 to the power of 2. since 9 is 3 times 3, so 3 squared raised to the power of 4 over 3 is equal to and 27 is the same as 3 times 3 times 3, which is 3 to the power of 3 raised to the power of 8. divided by 9, now we have 3 to the power of 2 times 4, giving us 8 divided by 3 is equal to 3 Divide itself to give us 1, divide 9 to give us 3, so this side is also 3 to the power of 8 over 3. The left-hand side is equal to the right-hand side. Thank you so much for watching. Bye.