Can you find the bigger number without a calculator?

You're welcome back to my channel. This particular question is going to surprise you. Now, we are asked to find which of these values is greater than the other. So, are we going to have greater here or less than here? So, 50 to the power of 50 and 49 to the power of 51 without the use of a calculator, which is greater? First of all, let's divide both sides of this equation by 49 to the power of 51 so if we divide this so we have 50 to the power of 50 divided by 49 to the power of 51 and then we divide this side by the same thing okay so now we have 50 to the power of 50 divided by 49 to the power of 51 and this will give us one we are going to concentrate on the left side of this equation, okay? Let's do that. We can express 51 as 50 plus 1, okay? 51 is equal to 50 plus 1. So, that means that we can rewrite this equation as 50 to the power of 50 divided by 49 to the power of 50 plus 1. Now, remember that a to the power of m multiplied by a to the power of n is equal to a to the power of m plus n using the law of exponents. And that simply means that we can express this denominator as 50 to the power of 50 divided by 49 to the power of 50, right? then multiplied by 49 to the power of one if this sorry about that if this is the first time you have seen this channel please click the subscription button turn on your notification bell so that you will be notified anytime i post a new video please drop your comment in the comment section let me know how you feel about the video things i need to improve on and the things i did okay and also if you have a shorter way of solving this particular equation please go ahead and do so and tell us show us in the comment section as well now moving forward we can now rewrite this equation as 50 raised to the power of 50 divided by 49 raised to the power of 50 multiplied by 1 over 49 to the power of 1 is still 49 so i'll leave it like this now recall that a to the power of m divided by b to the power of m is equal to a divided by b raised to the power of m so that means you can write this as 50 to the power of 50 50 divided by 49 raised to the power of 50 multiplied by 1 over 49 moving forward Someone said in the comment section the other time that i used to say a lot of moving forward Of course, we keep moving forward until we get to the last part of the equation Okay, now look at this We have 50 over 49 raised to the power of 50 multiplied by 1 over 49 So we can express 50 as 49 plus 1 We are heading somewhere. Remember, we are not allowed to use calculator. so we use our brain so we have 49 plus 1 divided by 49 raised to the power of 50 multiplied by 1 over 49 all right so um we can rewrite this equation we can write this equation as i don't know why i'm minimizing okay let's continue so we can write this equation as 49 or this fraction as 49 over 49 plus 1 over 49. This particular part raised to the power of 50 then multiplied by 1 over 49. Yes, because if we look at this bracket, we can write this as 49 plus 1, which is 50, right, over 49 as well. So we have not changed anything. Therefore, if we should factorize this further, if we should factorize this further or simplify this further we are going to have 49 divided by 49 is 1 plus 1 over 49 raised to the power of 50 multiplied by 1 over 49 so this will be equal to I'm looking at something recall that the limit of 1 plus 1 over n raised to the power of n as n tends to infinity is equal to e and that e is 2.7 all right so e is actually less than 3. Now let me see if there is a way I can actually express this in this form so and there is a way to do that so I can write this as 1 plus 1 over 49 raised to the power of 49 plus 1 okay then multiplied by 1 over 49 again so if we should expand this we are going to have this is our base 1 plus 1 over 49 raised to the power of 49 multiplied by 1 over okay multiplied by 1 plus 1 over 49 that is the base raised to the power of 1 then multiply by 1 over 49 this is what this is equal to at this point i hope you are enjoying this video okay now let's continue looking at this we can see that this is already in the form we talked about that means that this is what less than 3 so we can write that 1 over 1 plus 1 over 49 raised to the power of 49 is less than three then multiplied by this right now this is same thing as 50 over 49 50 over 49 then multiplied by 1 over 49 so let's also multiply this side by 50 over 49 multiplied by 1 over 49 because this exactly represents what we are working on for the left hand side so we are going to have 50 over 49 squared into 1 plus 1 over 49 right raise to the power of 49 and this will be less than 3 multiplied by 50 over 49 squared all right there are so many ways to solve this but this method i'm using so now we have 50 over 49 squared bracket 1 plus 1 over 49 raised to the power of 49 is less than this is 150 divided by 49 squared now you can see clearly we are going to have a number that is less than 1 here that is because 49 squared is going to give us a huge number is bigger than 150 right so 150 divided by any number that is bigger than, it's going to give us a smaller number, a number that is less than 1. So we can say that the whole of this is less than 1. And remember that we are trying to solve it together, or we are trying to compare it with this side after dividing both sides by this. That means we are comparing it with 1. And if this side is less than 1, then it means that 50 to the power of 50 is less than 49 to the power of 51 thank you so much for watching and see you in my next video bye