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[00:00:00]Uh, like before, we will work on two GMAT quant problems today. And uh I'll get right down to it.

[00:00:12]So, the first one is a problem uh consisting of some numbers. It's square root of Inside the square root, there's another square root, 1 + 17 / And we are required to find the value final value of this expression. So, we will work it out in steps like we always do.

[00:00:35]So, we can write this as square root of square root of For one, we can write 64 over 64 + 17 over 64.

[00:00:49]Why did I write the one as 64 over 64?

[00:00:54]The reason is simple. Uh I want to keep the denominators in both the terms the same. The denominator in the second term was 64, hence I kept the denominator in the first term also 64. But, the first term is actually one. So, 64 / 64 is actually one. That's why I put 64 in the numerator as well as in the denominator. So, one is multiplied by 64 in the numerator, one is multiplied by 64 in the denominator.

[00:01:26]Which finally simplifies to square root. Now, 64 + 17 is the 81.

[00:01:35]7 + 4 is 11, carry the one. 6 + 1 is 7 + 1 is 81 over 64.

[00:01:42]Okay, so And that we can write as square root of Now, the square root of 81 is, as you know, 81 is 9 squared and 64 is 8 squared.

[00:01:58]So, this will become equal to square root of 9 squared is 9 and square root of 8 squared is 8 inside the outer square root.

[00:02:10]Now, for 9, we can write as 3 squared. 3 * 3 is 9.

[00:02:18]So, let's write it like this. 3 * 3 and for 8, we can write as 2 * 2 * 2.

[00:02:29]Okay?

[00:02:31]So, we can write this as square root of 3 squared divided by 2 squared multiplied by 2. The first 2 is 2 to the power of 2 2. Now, the square root of 3 squared is 3.

[00:02:49]Square root of 2 squared is 2.

[00:02:51]And then, we are left with a square root of 2 in the denominator. But, in algebra, we would like to rationalize the denominator. That means we do not want to leave a square root here in the denominator. So, since there is a square root of 2 already in the denominator, we will multiply the denominator by square root of 2. Also, the numerator, we will multiply by 2 square root of 2. The reason is this keeps the fraction square root of 2 divided by square root of 2 as 1.

[00:03:23]So, this now becomes 3 square root of 2 in the numerator divided by 2 and inside the square root of we have 2 multiplied by 2.

[00:03:36]Or, 3 square root of 2 divided by 2 and square root of 2 squared itself is 2.

[00:03:45]So, the final answer is 3 divided by 4 square root of 2.

[00:03:52]That is the final simplified form.

[00:03:54]That's your answer.

[00:03:56]I will go over it in steps once I'm done with the second problem.

[00:04:00]Okay.

[00:04:02]So, let's take a look at the second problem.

[00:04:06]A group of 200 friends formed an investment club, each member contributing an equal amount. The club then invested the entire amount of D dollars. D was the total investment. In stock X, the value of the stock increased 40% when it was sold and the proceeds divided equally among the members. In terms of D, how much money did each member receive?

[00:04:40]Okay, after the stocks were sold. So, they invested invested D dollars in stock X which appreciated or which appreciated 40%.

[00:05:07]Appreciated means increased in price by 40%.

[00:05:14]by 40%.

[00:05:20]Or 0.4 D.

[00:05:28]So, when the stock was sold, the value was D, original amount plus 0.4D.

[00:05:53]Now, as you know in algebra, D means 1 * D.

[00:05:57]So, in algebra, you add the coefficients, so that becomes 1.4 D.

[00:06:06]Okay?

[00:06:07]>> [clears throat] >> Now, 1.4D was divided equally among 20 members, so each member received received Let's just see.

[00:06:30]is R E C E I V 1.4D divided by 20.

[00:06:45]Now, if you take the common factors out, 1.4 is 2 multiplied by 0.7 D divided by 2 * 10.

[00:06:58]So, this two and this two cancel.

[00:07:01]So, you are left with 0.7 by 10 D.

[00:07:11]Okay? Or you can simplify this by multiplying the numerator and denominator by 10.

[00:07:21]* 10 / 10 * 10.

[00:07:28]So, when you multiply by numerator by 10, you have to multiply the denominator by 10.

[00:07:32]Why did you multiply the numerator by 10 and denominator by 10? 10 / 10 is 1.

[00:07:38]Now, 0.7 * 10 is 7.

[00:07:41]So, you have 7 over 100 10 * 10 is 100 D as what each person received after the sale of the stock.

[00:07:52]So, let's walk again step-by-step on both the problems. So, the first problem is square root and inside the square root there's another square root of 1 + 17 over 64.

[00:08:05]And you have been asked to find the value of this expression.

[00:08:09]Now, uh So, we will work with inside square root first. Inside the second square root inside is 1 + 17 over 64. 1 is expressed as 64 over 64.

[00:08:22]And leave the 17 by 64. Why 64 over 64 for 1? Because 64 then becomes a common denominator in both. So, 64 and 64 common denominator in both. 64 + 17, if you do the addition, 7 + 4 is 11. Write the 1.

[00:08:41]Carry the 1 because it's 11. 6 + 1 is 7 + the carried 1 is 8. So, it's 81 over 64. Now, 81 and 64 are both perfect square. 81 being 9 squared, 64 being 8 squared. So, inside the square root you have 9 squared by 8 squared.

[00:09:01]And square root of 9 squared and square root of 8 squared is 9.

[00:09:05]And then square root of 8 squared is 8.

[00:09:07]So, it becomes square root of 9 over 8.

[00:09:10]Inside square root now gone.

[00:09:12]Now, 9 is 3 squared.

[00:09:15]8 is 2 * 2 * 2. So, it's 2 squared * 2.

[00:09:20]So, we have square root of 3 squared is 3 and square root of 2 squared in the denominator is 2 and then we will be left with an extra square root of 2 in the denominator.

[00:09:33]In algebra, we do not leave a square root in the denominator. So, we try to rationalize it by multiplying by that square root that is left in the denominator. Square root of two in the denominator and square root of two.

[00:09:47]Remember when you multiply the denominator by a number, you have to multiply the numerator at the top by the same number, too. Why? Because these two divided by is one. Okay. So, in the numerator, you have three square root of two. In the denominator, you have two and inside the square root, you have two times two, which is four. Whose square root is two. So, in the numerator, you have three square root of two divided by two times two is four. So, the final answer is three divided by four square root of two. That's your answer.

[00:10:22]For the first problem.

[00:10:24]Any question you have, you can always shoot me a question in the comments area.

[00:10:29]Okay.

[00:10:40]Okay. Let's look at the second problem.

[00:10:44]A group of 20 friends formed an investment club with each member contributing an equal amount.

[00:10:50]The club then invested the entire amount of D dollars in stock X.

[00:10:55]The value of the stock increased 40% when it was sold and the proceeds divided equally among the members.

[00:11:07]In terms of D, how much money did each member receive?

[00:11:11]So, the investment is D dollars.

[00:11:14]It appreciated by 40% or 0.4D.

[00:11:18]So, the stock value finally became D plus 1.4D.

[00:11:25]1D plus 0.4D.

[00:11:28]And then it becomes when When add the coefficients is 1.4d.

[00:11:33]Now you divide it into 20 equal parts.

[00:11:35]So you divide it by 20. So you have 1.4d divided by 20 is what each person received. Which you can simplify further. For 1.4 you can write 2 * 0.7.

[00:11:48]For 20 in the denominator we can write 2 * 10. The idea is to cancel the common factor in the numerator and denominator which is 2.

[00:11:57]So you're left with after canceling the 2 in the numerator and denominator 0.7d over 10.

[00:12:04]Now you can further simplify this by multiplying the numerator and denominator by the same number 10 over 10. Why do you do that? Because 0.7 * 10 now becomes 7. In the denominator you have 10 * 10 is 100. So the final answer what each person receives is 7 / 100 * d. 7 over 100 * d.

[00:12:29]That's what each person receives.

[00:12:34]Okay, I will stop here today. You may have some questions, no problem. You can always shoot me a question in the comments area. If you like my video, please give me thumbs up.

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[00:12:49]And if you like, please share this with your friends.

[00:12:52]And please do not forget to check me out tomorrow when I'll be back with another interesting problem. Take care. Have a nice day and see you. Bye.