Harvard University Interview Tricks.

[00:00:00]Hello. You're welcome to solve this math problem, which is 7 ^ t + 7 ^ 2t + 7 ^ 3t is equal to 14.

[00:00:12]To find the value of t from this equation such that t is element of a real number.

[00:00:19]Now, from this our problem here, which is 7 ^ t + 7 ^ 2t is same as 7 ^ 2 * t.

[00:00:30]Then plus 7 ^ 3t is same as 7 ^ 3 * t is equal to 14.

[00:00:39]Then from this part here and this part is in the form of the rule of a ^ m * n, which is equal to a ^ m inside the bracket ^ n outside the bracket.

[00:00:55]Then we apply this form here.

[00:00:57]So, into here, it will be 7 ^ t. We'll make this 7 ^ t in common. Plus here it will be 7 ^ t inside the bracket.

[00:01:09]Then ^ 2 outside the bracket. Plus 7 ^ t inside the bracket.

[00:01:17]^ 3 outside the bracket is equal to 14.

[00:01:23]Then in the next step, here 7 ^ t is common. So, we let u is equal to 7 ^ t.

[00:01:32]So, into here, it will be u plus u squared.

[00:01:38]Then plus u ^ 3 is equal to 14.

[00:01:45]Then here we'll start with the largest power. So, it will be u ^ 3 plus this u squared plus this u. We'll take 14 to this side, so it'll be minus 14 is equal to zero.

[00:02:01]Then into here, it'll be U ^ 3 + U ^ 2 + U 14 we split into addition of numbers which can be in form of exponents. So, 14 is same as 8 + 4 is 12 + 2 is 14 is equal to zero.

[00:02:27]>> [snorts] >> Then here, it'll be this here we change into exponents, so it'll be U ^ 3 + U ^ 2 + U. Here we take negative inside, so it'll be minus 8 into exponent is 2 ^ 3.

[00:02:43]Here it'll be minus 4 into exponent is 2 ^ 2.

[00:02:47]Here it'll be minus 2 is equal to zero.

[00:02:52]Then here, it'll be we stack powers of 3 together, so U ^ 3 then minus 2 ^ 3 then plus this U ^ 2 here minus 2 ^ 2 then here plus U minus 2 is equal to zero.

[00:03:14]Then from this part here, this part we'll apply a rule which is in form of a ^ 3 - b ^ 3 is equal to a - b bracket bracket a ^ 2 + ab + b ^ 2 bracket.

[00:03:38]So, here we apply this form. A - b it'll be U - 2, so here it is U - 2 bracket bracket a ^ 2 it'll be U ^ 2. Plus ab it'll be U * 2. is two u plus b squared, it is two squared.

[00:04:00]Then bracket then plus this here u squared minus two squared plus u minus two is equal to zero.

[00:04:13]Then in the next step from this part here into here we'll apply rule which is here we'll apply rule which is a squared minus b squared is equal to here a minus b bracket bracket a plus b bracket. So we'll apply this form into here.

[00:04:38]Then it will be u minus two bracket bracket u squared plus two u plus two squared, it is four bracket then plus into here is in this form then we change into this form. A minus b, it will be u minus two, so u minus two bracket bracket a plus b, it will be u plus two bracket then plus this here u minus two bracket is equal to zero.

[00:05:12]Then from here u minus two is common, so take u minus two bracket outside the bracket. Now this divide by this, it is this here u squared plus two u plus four plus this divide by this here u plus two plus this divide by this, it is one bracket is equal to zero.

[00:05:42]Then into here it will be u minus two bracket bracket here you square plus two you plus you it is three you plus four plus two is six plus one is seven bracket is equal to zero.

[00:06:00]Then from here we have two solutions.

[00:06:03]Well, this first solution of you minus two is equal to zero and this second solution of you square plus three you plus seven is equal to zero.

[00:06:16]Then from this first solution we take negative two to this side so it will be you is equal to two.

[00:06:23]Here we got you. Then here this is real solution.

[00:06:28]Here let's check if it will give a real solution. So we'll use discriminant formula which is equal to b square minus four ac.

[00:06:38]Where as coefficients a is equal to coefficient of you square it is one. B is equal to coefficient of you it is three and c is equal to constant is seven. So it will be discriminant is equal to b square it is three square minus four times a it is one times c is seven.

[00:06:59]So it will be discriminant is equal to three square it is nine minus four times seven is 28.

[00:07:09]Then into here it will be discriminant is equal to nine minus 28 is minus 17. Now this value is less than zero. So into here no real solution.

[00:07:28]Then we'll continue from the solve from this first solution. Where as in the first we let you is equal to seven power of t.

[00:07:37]So here we'll use you is equal to seven power of t.

[00:07:43]So, we'll compare here. It will be 7 power of t is equal to 2.

[00:07:49]Then, we apply log in both sides. So, it will be log of 7 power of t is equal to log of 2.

[00:07:59]Then, it will take power of t in the base. So, it will be t log of 7 is equal to log of 2.

[00:08:10]Then, we divide by log of 7 in both sides. So, here it will be t log of 7 is equal to log of 2.

[00:08:22]Then, here we divide by log of 7 and here we divide by log of 7. So, this and this will cancel. Then, it will be t is equal to log of 7 divide log of 2 divide by log of 7 is same as log of 2 base of 7.

[00:08:42]So, this is the final answer for the value of t into this our problem.

[00:08:49]Now, in the next step, let's check this answer if it is correct.

[00:08:53]Now, to check from t is equal to log of 2 base of 7.

[00:09:01]From our problem here, which is 7 power of t plus 7 power of 2t plus 7 power of 3t is equal to 14.

[00:09:13]Then, into here we substitute our answer here. So, it will be 7 power of t. So, it will be power of log of 2 base of 7 plus 7 power of t 2t it will be 2 * t it is log of 2 base of 7 plus 7 power of 3t it is 3 * log of 2 base of 7 is it equal to this 14?

[00:09:46]Then into here, it will be 7 power of log of 2 base of 7 + 7 Here we take this two in the power, so it will be log of 2 power of 2 base of 7 + 7 Here we take three in the power, so log of 2 power of 3 base of 7 is it equal to 14?

[00:10:14]Then into here, it will be 7 power of log of 2 base of 7 + 7 power of log of 2 power of 2, it is 4 base of 7 + 7 log of 2 power of 3, it is 8 base of 7 is it equal to 14?

[00:10:35]Then from this part, this part, this part is in the form of a power of log of b base of a, which is equal to b.

[00:10:47]So, we'll apply this rule here.

[00:10:49]So, from 7 power of log of 2 base of 7 is same as 2.

[00:10:54]Plus 7 power of log of 4 base of 7, this here is same as 4.

[00:11:00]Plus 7 power of log of 8 base of 7, this here is same as 8.

[00:11:06]Then is it equal to 14?

[00:11:10]Now, 2 + 4 is 6, 6 + 8 is 14 is equal to 14.

[00:11:17]So, left side and right side are equal.

[00:11:20]Then it is true for the value of t is log of 2 base of 7 which is the only real solution.

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[00:11:39]Bye-bye.