Harvard University Interview Tricks.

Added: 2026-05-19

[00:00:00]Hello, you are welcome to solve this math problem which is square root of 12 over 15 + 3 bracket^ of 12. Now solution into here it will be equal to from square root of 12 is same as square root 12 is same as 3 * 4 then over here square root of 15 is same as square root 15 is same as 3 * 5 then plus this square root of 3 bracket this power here power of 12 then in The next step here it will be equal to from here we separate so it will be square<unk> of 3 * square<unk> of 4 over also into here we separate the square roots so it will be square<unk> of 3 *<unk> of 5 then plus this square<unk> of 3 bracket this power here power of 12 then in the next step here it will = to square<unk> of 3 *<unk> 4 it is 2 then over in the denominator roo<unk> of 3 is common so take roo<unk> 3 outside the bracket now 3 * 5 / 3 it is this square roo<unk> of 5 plus 3 / 3 it is 1 bracket then bracket this power here power of 12.

[00:01:44]Then in the next step here we simplify this square root of three. You cancel this square root of three. So it will be equal to this here which is 2 over<unk> of 5 + 1 bracket^ [snorts] of 12.

[00:02:01]Then into here it will be equal to 2 over here the denominator. So here square<unk> of 5 + 1 to rationalize the denominator we multiply by square roo<unk> of 5 here it is + one here to be min -1 bracket also into the numerator we multiply by this square roo<unk> of 5 -1 bracket then bracket this power here power of 12 then into here it will be equal to this time this so here 2 bracket 5 - 1 bracket then over here 5 + 1 * 5 - 1 is same as square<unk> of 5² - 1 square then bracket this power here power of 12 then into here it will be equal to 2 bracket square<unk> of 5 - 1 bracket then over here square root of 5 square this square you cancel square so it will be 5 - 1 square it is 1 bracket [snorts] this power here power of 12 then in the next step here it will be 2 bracket square<unk> of 5 - 1 bracket over 5 - 1 it is 4 then bracket this part here power of 12. Now into here to simplify this two cancel by 1 this by 2.

[00:03:48]So it will be equal to square<unk> of 5 - 1 / 2 then bracket^ of 12 then from here this here inside the bracket we let as x. So here we let xc is equal to this here which is roo<unk> of 5 - 1 / 2. So here if this will let us x we'll be finding x c^ of 12.

[00:04:25]Then in the next step here we will cross multiply. So it will be x * 2 it is 2 x is equal to this<unk> of 5 - 1. Then we take this -1 to the left side. So it will be 2 x + 1 is =<unk> of 5.

[00:04:49]Then into here we square in both sides.

[00:04:54]So it will be 2 x + 1 bracket² is equal to this square roo<unk> of 5 bracket².

[00:05:05]So to expand this here it will be 2 x bracket square + 1² + 2 * this 2x * this 1 is equal to here this square you can say square. So is equal to 5. Then here 2x bracket square. So 2 square is 4 x² + 1² is 1 + 2 * 2x is 4 x * 1. Here it is 4 x is = 5.

[00:05:43]Then into here we'll take this one to this side. So here it will be 4 x² is = 5 1 take this side will be -1 and 4x we take this side it will be - 4x then here it will be 4 x² is = 5 - 1 it is 4 - 4x then here it will be 4 x² is = here 4 is common. So we take 4 outside the bracket. 4 / 4 is 1 - 4x / 4 is - x bracket. Then into here we divide by four in both sides. So this and this will cancel. Then it will be x² is equal to this and this will cancel. So it will be equal to 1 - x. Then note this value of x² which is 1 - x. Now here it is x² until we make x^ 12.

[00:06:58]So here x² let's x² we square again in both sides. So it will be 1 - x bracket also into this side we square. So it will be x e^ 4 is = 1² then + x² then - 2 * 1 * x then it will be x e^ 4 is = here 1 2 is 1 then plus this x² - 2 * x is - 2 X then into here it will be X E^ 4 is equal to here 1 - 2 1 - 2 XC then form plus here X² we substitute here 1 - X so here 1 - X then into here it will be X^ 4 is equal to here 1 + 1 it is 2 - 2x - x is - 3 x then here x^ 4 until x^ 12 we will power by 3 in both sides so it will be x^ 4 bracket here by 3 is = 2 - 3 x bracket also into here by 3 then into here it will be x c^ of 12 4 * 3 is 12 is equal to here the expansion of this this here it is in the form of a - bracket^ 3 which is equal to a^ 3 - 3 a² b then plus 3 a b² then minus b^ of 3. So we'll apply this form here.

[00:09:06]Then here 2 is a 3x is b. So a^ 3 it will be 2^ 3. Then - 3 * a² it will be this 2² then * b is this 3 x then plus 3 a a it is 2. Then time b² it will be 3 x bracket² then minus b^ 3 it will be this 3 x bracket^ 3 then into here it will be x c^ of 12 is equal to 2^ 3 it is 8 then minus here it will 3 * 2^ 2 it is 4. So here it is 3 * 4 it is 12. Then * this 3 x then + 3 * 2 is 6. Then * 3² it is 9 x² then - 3^ 3 is 27 x^ 3.

[00:10:25]Then into here it will be x e^ of 12 is = 8 - 12 * 3 is 36 x then plus into here 6 * 9 is 54 x² then - 27 x^ 3 we make into x² so x^ 3 is same as x² * x. So as in the later step into x² we will substitute 1 - x.

[00:11:01]So here it will be x c^ of 12 is = 8 - 36 xc then + 54 x² we substitute x² which is 1 - x bracket - 27 x² we substute 1 - x bracket then times x Then here it will be x e^ of 12 is = 8 - 36 x c. Then here it will be + 54 * 1 is 54.

[00:11:45]54 * -x [clears throat] is - 54 x - 27 * 1 is - 27 * x is - 27 x - 27 * -x is 27 x * x is positive 27 x² then here it will be X C^ of 12 is = 8 - 36 XC + 54 - 54 XC - 27 XC then + 27 into X² we substitute it is 1 - X so here it will be X^ 12 is equal to 8 + 54 it is 62.

[00:12:52]Then here - 36x - 54x here it will be - 6 + 4 it is 10.

[00:13:04]So we go with one 3 + 5 is 8 + 1 it is 9 x then into here this here - 27 xc then here plus this * this is 27 27 * -x is - 27 x then into here it will be x e^ of 12 is equal to Here it will be 62 + 27. Now 2 + 7 is 9. Then 6 + 2 it is 8. Then here this plus this.

[00:13:51]So it will be minus here 0 + 7 is 7. 9 + 2 is 11 x. Then we have this here. So - 27 x then here to be x e^ of 12 is = 89.

[00:14:10]Then here it will be minus 7 + 7 is 14 go with 1. 1 + 2 is 3 + 1 it is 4. Then here 1 x then here it will be x e^ of 12 is = 8 9us here we get x^ 12 then -1 4 x into x we recall before we let x it is this here 5 - 1 / 2 so here bracket square<unk> of 5 - 1 bracket over 2. So here it will be x c^ of 12 is equal to 89. Then minus here to simplify this by 1 this by 72.

[00:15:08]So here it will be 72 bracket this here roo<unk> of 5 - 1 bracket.

[00:15:16]So here it will be x c^ of 12 is = 89. Then -72 *<unk> 5 is -72<unk> 5 - 72 * -1 is + 72.

[00:15:37]Then into here it will be x e^ of 12 is equal to here we add. Now 9 + 2 it is 11 go with 1. Then 8 8 + 7 is 15. 15 + 1 it is 16.

[00:15:58]Then this here - 72<unk> of 5. So this here the simplified form of square<unk> of 12 over square<unk> of 15 -<unk> of 15 square<unk> of 15 +<unk> 3 bracket bar of 12 is equal to is equal to this here which is 161 - 72 square<unk> of 5. So this is our final answer.

[00:16:44]Thank you for watching. Don't forget these tips to miss out. Subscribe to my channel and see you in the next video.

[00:16:54]Bye-bye.