Solving a 'Stanford' University entrance exam ?

Hello everyone how to simplify this nice algebraic expression without using the calculator if we have 60 to the power of 6 minus 50 to the power of 6. So we write this algebraic expression into the simplified form and we move towards the solution of this problem. I hope so you like this method but if you have any other method in your mind so please don't hesitate to write this method and here in this method first we rewrite this equation as this is 60 to the power of 6 and we write this 6 as 3 multiplied by 2 okay minus this is 50 to the power of 6 and we write it as 2 multiplied by 3. So here further we use the nice exponential law you know about that if we have a to the power of m multiplied by n we write it as a to the power of m to the power of n. According to this rule of exponents here we move this 3 inside the term and it will be written as 60 cubed and its whole square minus this will be written as 50 cubed and its whole square and now here in the next step we use the nice algebraic formula and you know about that if we have a square minus b square is equals to a minus b times of a plus b and you see here the base of this term is our a and the base of this term is our b. So according to this formula we write our above expression as in this form. So here we have our a is 60 cubed minus b is 50 cubed into again our a is 60 cubed plus b is 50 cubed and further in the next step here we need to find the cube of 50 and 60. So this is our challenge because calculator is not allowed in this problem so we need to do all of the working on this paper. So here first we will find the value of 60 cubed and you very well know about that 60 cubed is means that we multiply 60 with 3 times okay. So here first we will take that as 60 cubed. So here first we multiply these two values and here you consider this is our one digit number and this is our two digit numbers. So here we first multiply 0 with this complete number 60 we get 0 times 60 becomes 0 and we multiply 6 with this number you get 6 times 0 becomes 0 and 6 x 6 are 36 and here our resulting answer is multiplied by the 60 okay. So we replace this value here and now again we consider this is one digit number and this is our two digit number. So here we multiply 0 with this 300 3600 so it is equals to 0 and 6 times this number is equals to 6 times 0 becomes 0 6 0s are 0 6 6s are 36 we write 6 and carry 3 6 3s are 18 18 plus 3 becomes 21. So this is 60 cubed and in the same pattern we need to find 50 cubed. So here and now we will just take here 50 cubed and you know about that the meaning of 50 cube is that we multiply 50 with 53 times in this way okay. And now here again we multiply first these two values and again we consider this is one digit number and this is two digit number. So 0 times this number equals to 0 5 times this number equals to 5 0s are 0 and 5 5s are 25 and our resulting answer is multiplied by this 50. And in the next step again we consider this is one digit number and this is our two digit number so 0 times of this number equals 0 5 times this number equals to 5 0s are 0 5 0s are 0 5 5s are 25 we write 5 and carry 2 5 2s are 10 10 plus 2 becomes here 12. So here we need to replace this value and this value into this expression okay. Our this expression is written as in this form. So here we first copy down our above expression it is 60 to the power of 6 minus 50 to the power of 6 is equals to 60 cubed minus 50 cubed multiplied by 60 cubed plus 50 cubed okay. So this is our above expression and now we substitute above both of these two values here. So the value of 60 cubed is 2 1 6 and triple 0 minus the value of 50 cube is 1 2 5 and triple 0. Then similarly we substitute it here and it will becomes 2 1 6 triple 0 plus 1 2 5 and triple 0 okay. We substitute above both of values here. So in the next step we need to subtract these two values so 0 minus 0 becomes here 0 this 0 minus 0 again becomes 0 this 0 minus 0 becomes 0 6 minus 5 becomes 1 and here 1 is not subtracted by 2. So here we carry 1 from here this will becomes 1 and this will becomes 11. So 11 minus 2 becomes 9 and this 1 minus 1 becomes 0. So this will be written as 91 000 and it is multiplied by here we need to sum up both of these values and it will becomes 0 plus 0 becomes 0 this 0 plus 0 is 0 this 0 plus 0 is 0 6 plus 5 becomes 11 we write 1 and carry 1. So this 1 plus 2 is 3, 3 plus 1 is 4 and 2 plus 1 is 3. So this is our 3 4 1 and triple 0. So here our next order is to multiply these two bigger values without using the calculator okay. So here we need to break this value and we write this value as it is this is 3 4 1 and triple 0 is multiplied by we break this value as 90 000 and plus this is 1000 okay. So here when we multiply this complete number with this value so we again consider this is one digit number and this is our five digit number. So here we need to just first write these four zeros here this is 0 0 0 and 0 and now we multiply 9 with this value it will becomes 9 0 0 9 0 0 9 0 0 and 9 times 1 becomes 9, 9 4s are 36 we write 6 and carry 3, 9 3s are 27, 27 plus 3 becomes 30. So it will becomes here in this form and when we multiply this number with 1000 we just write these three zeros here and it will becomes 3 4 1 triple 0 and these are three zeros okay. So here in the final step we need to sum up both of these values. So here we have these are 2 4 and 6 zeros these are the 2 4 and 6 zeros. So when you add these six zeros it will becomes here six zeros okay. So these are 2 4 and 6 and we as we just add the remaining value. So this is 0 plus 1 becomes 1 this is 9 plus 4 is equals to 13 we write 3 and carry 1. So this is 6 plus 3 becomes 9, 9 plus 1 becomes 10. So this is 10 and we get here this is 3 only okay. So this is our final simplified form of the given question and this is the final answer and thank you so much for watching this video. Please subscribe to my channel for more exciting videos.