Harvard University Interview Tricks

Added: 2026-05-08

[00:00:00]Hello. You're welcome to solve this math problem of simplifying square root of 18 plus square root of 12 plus square root of 6 over square root of 18 minus square root of 12 plus square root of 6 bracket this power of 8.

[00:00:19]So, here it will be equal to from square root of 18 to have square root of 6, this here is same as square root of 18 is same as 6 * 3. Then plus square root of 12 is same as square root of 6 * 2. Then plus this square root of 6.

[00:00:40]Then over into here square root of 18 is same as square root of 6 * 3.

[00:00:49]Then minus square root of 12 is same as square root of 6 * 2. Then plus this square root of 6.

[00:00:59]Bracket this power of 8.

[00:01:03]Then in the next step it will be equal to into here we separate the square root.

[00:01:08]So, this here is same as square root of 6 * square root of 3 plus here we separate it will be square root of 6 * square root of 2. Then plus this square root of 6.

[00:01:23]Over into here we separate so it will be square root of 6 * square root of 3.

[00:01:31]Then minus here we separate it will be square root of 6 * square root of 2.

[00:01:39]Then plus this square root of 6. Bracket this power of 8.

[00:01:47]Then in the next step here it will be equal to into here square root of 6 it is common.

[00:01:53]So, we will take square root of 6 outside the bracket. This here divided by square root of 6 it is this square root of 3. Then plus this here divided by square root of 6 it is this square root of 2. Then plus square root of 6 divided by square root of 6 it is 1. Bracket then over in the denominator here square root of 6 is also in common. So, we take square root of 6 outside the bracket.

[00:02:24]This divided by square root of 6 it is this square root of 3.

[00:02:29]Then minus this divided by square root of 6 it is square root of 2. Plus square root of 6 divided by square root of 6 it is 1.

[00:02:38]Bracket then bracket here outside it is power of 8.

[00:02:46]Then here it will be equal to here we simplify square root of 6 will cancel square root of 6. So, it will be this here which is square root of 3 plus square root of 2 plus 1 over this which is square root of 3 minus square root of 2 plus 1 bracket this power of 8.

[00:03:14]Then in the next step here it will be equal to we rearrange so we take this plus this so square root of 3 plus 1 because it is positive. Then like this square root of 3 plus 1 also in the denominator.

[00:03:30]Then here it will be plus in the numerator plus square root of 2.

[00:03:35]Then over in the denominator it will be square root of 3 plus 1. So, here square root of 3 plus 1.

[00:03:44]Now, this is same as this but in the denominator we have minus square root of 2. Then bracket this power of 8.

[00:03:56]Then in the next step here because this here we have minus square root of 2 we rationalize the denominator.

[00:04:03]So, it will be in the numerator it is square root of 3 plus 1 plus square root of 2 over in the denominator it is square root of 3 plus 1 minus square root of 2.

[00:04:24]Now, here we rationalize the denominator. So, we will multiply by square root of 3 plus 1. Here it is minus square root of 2 so here it will be plus square root of 2. Bracket here bracket. Here we multiply by the same in the numerator so it will be square root of 3 plus 1 plus square root of 2. Bracket bracket this power of 8.

[00:04:52]Then into here it will be equal to now this times this they are the same so it will be square root of 3 plus 1 plus square root of 2. Bracket square.

[00:05:09]Then here this part times this here this here square root of 3 plus 1 is in the form of minus square root of 2 is in the form of a minus b. Bracket here square root of 3 plus 1 plus square root of 2 is in the form of a plus b.

[00:05:30]Bracket which is equal to a square minus b square. So, we apply this form.

[00:05:39]Then it will be this here which is square root of 3 plus 1. Bracket square minus square root of 2 square. Then bracket this power here power of 8.

[00:05:59]Then in the next step here >> [snorts] >> it will be it will be equal to this which is square root of 3 plus 1 plus square root of 2. Then here bracket square then over here in the denominator we expand this.

[00:06:22]This part square root of 3 plus 1 bracket square is in the form of a plus b bracket square which is equal to a square plus b square plus 2 a b. So, we apply this form.

[00:06:41]Then it will into here it will be here square root of 3 square plus 1 square plus 2 times this times this times square root of 3 times 1.

[00:06:56]Then minus square root of 2 square. This square root will cancel square so it will be minus 2.

[00:07:05]Then here bracket this power here power of 8.

[00:07:12]Then in the next step here it will be equal to square root of 3 plus 1 plus square root of 2. Bracket square over in the denominator here this will cancel so this way it will be 3 plus 1 square it is 1. Then plus 2 times 1 is 2 times square root of 3 it is 2 square root of 3.

[00:07:40]Then minus 2. Then here bracket power of 8.

[00:07:48]Then in the next step here it will be equal to this which is square root of 3 plus 1 plus square root of 2. Bracket square over into here 3 plus 1 is 4 minus 2 is 2.

[00:08:09]So, it will be 2 plus this 2 square root of 3. Then bracket this power of 8.

[00:08:19]Then in the next step it will be equal to from this part here square root of 3 plus 1 plus square root of 2 bracket square this part is in the form of the rule which is a plus b plus c bracket square which is equal to the expansion of this it is a square plus b square plus c square then plus 2 bracket a b plus b c plus c a bracket.

[00:08:56]Now, here we apply this form. So, a square it will be square root of 3 square plus b square it will be 1 square plus c square it will be square root of 2 square.

[00:09:13]Then plus 2 bracket a b it will be square root of 3 times 1 which is square root of 3. Plus b c it will be 1 times square root of 2 it is square root of 2.

[00:09:29]Plus c a it is square root of 3 times square root of 2. So, here square root of 3 times square root of 2. Then bracket over here in the denominator it is 2 plus 2 square root of 3. Then here bracket this power here power of 8.

[00:09:56]Then in the next step here will be equal to into here this square root will cancel square, so it will be 3 + 1 squared it is 1 + square root of 3 this square root will cancel square, so it will be + 2 then + 2 bracket Here, it will be square root of 3 + square root of 2 + square root of 3 * square root of 2 it is square root of 6 bracket then over Here, it is 2 + 2 square root of 3 bracket >> [snorts] >> this power of 8 Then, the next step here it will be equal to We add this 3 + 1 is 4 4 + 2 is 6. So, here it will be 6 + 2 bracket square root of 3 + square root of 2 + square root of 6 bracket then over this here 2 + 2 square root of 3 bracket this power of 8 Then, the next step here it will be equal to Here in the numerator 2 is common, so we take 2 outside the bracket. 6 / 2 it is 3 + here this 2 / 2 it is 1. So, it will be this inside the bracket, so + 3 square root of 3 + square root of 2 + square root of 6 bracket then over Here in the denominator 2 is common, so we take 2 outside the bracket. Here 2 / 2 it is 1 + 2 square root of 3 / 2 it is square root of 3 bracket then [snorts] bracket this power of 8 Then, into here this 2 will cancel this 2, so it will be equal to this here which is square root this which is 3 + square root of 3 + square root of 2 + square root of 6 over 1 + square root of 3 bracket this power of 8 Then, the next step here it will be equal to Into here 3 3 is same as square root of 3 * square root of 3 + this square root of 3 + this square root of 2 + square root of 6 is same as square root of 2 * square root of 3 then over 1 + square root of 3 then bracket this power of 8 Then, in the next step here it will be equal to Into here square root of 3 is common, so we take square root of 3 outside the bracket.

[00:13:19]This / square root of 3 it is square root of 3 + square root of 3 / square root of 3 it is 1 bracket then + Into here square root of 2 is common, so we take square root of 2 outside the bracket. This / this it is 1 + square root of 2 * square root of 3 / square root of 2 it is square root of 3 bracket then over this 1 + square root of 3 bracket this power of 8 Then, in the next step here it will be equal to Into here square root of 3 + 1 is common, so we take same as 1 + square root of 3 1 + square root of 3, so we take 1 + square root of 3 bracket outside the bracket.

[00:14:15]Now, this / this it is this square root of 3 + this / this it is this square root of 2 bracket then over this here 1 + square root of 3 bracket this power of 8 Then, the next step here To simplify this 1 + square root of 3 will cancel, so it will be equal to this square root of 3 + square root of 2 bracket this power of 8 Then, the next step from here We let this as X, so we'll be finding the value of X C power of 8 So, here we let X is equal to this inside the bracket which is square root of 3 + square root of 2 Now, until we make X power of 8 Here, it will be Until we make X power of 8 Here, it X we make into square, so it will be X squared in the first into the first step.

[00:15:40]is equal to Also, this will square, so it will be square root of 3 + square root of 2 bracket squared So, here it will be X squared is equal to Here to square this, it will be to the square of this it will be square root of 3 squared + square root of 2 squared + 2 * this * this, so * square root of 3 * square root of 2 So, here it will be X squared is equal to This square root will cancel square, so it will be 3 + Here square root will cancel square, so it will be + 2 + this 2 square root of 3 * 2 is equal to 6 So, here it will be X squared is equal to 3 + 2 is 5 + 2 square root of 6 Then, here X This is X squared.

[00:16:43]Not until we make X power of 8 So, here it will be X squared bracket squared is equal to Also, into this we square it, so it will be 5 + 2 square root of 6 bracket squared So, here it will be X power of 2 * power of 2 it is power of 4 is equal to Here we expand, it will be 5 squared + this 2 square root of 6 bracket squared + Here, it will be 2 * 5 * this 2 square root of 6 Then, here is it will be X power of 4 is equal to 5 squared is 25 + Here it will be 2 squared is 4 * square root of 6 squared it is 6 then + 2 * 2 is 4 * 5 it is 20 this square root of 6 Then, here it will be X power of 4 is equal to 25 + 4 * 6 is 24 + 20 square root of 6 Then, here it will be X power of 4 is equal to Into here 24 25 + 24 it is 49 then + 20 square root of 6 Now, here it is X power of 4 until to X power of 8 here it will be X power of 4 bracket squared is equal to Also, here we square it, so it will be 49 + 20 square root of 6 bracket squared Now, here it will be X C 4 * 2 is power of 8 is equal to Here, it will be 49 squared.

[00:18:52]Then, + this 20 square root of 6 bracket squared Then, + this here 2 * Sorry, 2 * this 49 * this 20 square root of 6 So, here it will be X power of 8 is equal to 49 squared.

[00:19:20]49 * 49 Here 9 * 9 is 81 goes 8 9 * 6 is 36 36 + 8 is 44 4 * 9 is 36 goes 3 4 * 4 is 16 + 3 it is 19 So, here it will be 1 here it is 10 Here it is 13 + 1 is 14 then here add 1 it is two.

[00:19:53]So, here 49 squared it is 2 4 0 1.

[00:19:58]Then, plus here 20 squared is 20 squared is 400 * square root of 6 squared it is 6. Then, plus into here 2 * 49 is 98.

[00:20:16]98 * 2 it is 196.

[00:20:22]Then, this zero * square root of 6.

[00:20:29]Then, into here it will be x x power of 8 is equal to this here, which is 2 4 0 1. Then, plus this here, 4 * 6 is 24. Then, 0 0 plus this here, which is 1 9 6 0 square root of 6.

[00:20:58]So, here it will be x power of 8 is equal to this plus this. Here, it will be 1 0 4 + 4 it is 8. 2 + 2 is 4. Then, plus this here 1 9 6 0 square root of 6.

[00:21:18]So, this here is the simplified form.

[00:21:22]So, our final answer it is equal to 4 8 0 1 plus this here, which is this, which is 1 9 6 0 square root of 6.

[00:21:41]So, this is our final answer.

[00:21:46]Thank you for watching.

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[00:21:55]Bye-bye.