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# GMAT: Math 38

190-194, pg. 178. Created by Sal Khan.

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• I don's understand in question 192 who -1^2 = -1. In my opinion it should be 1. Can somebody please explain? • I'm sure you have progressed through life just fine without having this question answered here, but someone else might wonder about it.

The question was to simplify the expression
(√2 + 1)(√2 - 1)(√3 + 1)(√3 - 1)

So you are asking about `-1²`
There is not any time during that part of the video where Sal squares -1
If he did, he would write it (-1)² and the result WOULD be positive 1. Meanwhile, the expression -1² always results in -1, due to PEMDAS. We do the square of the number that the exponent sits on, then we apply the negative multiplier in that expression.
If we read -1² the words are, "the negative of (1 squared)". It does not mean the square of -1, so if you ever want to convey "the square of -1" to someone, you should use the parentheses: (-1)²

Anyway, he was not squaring a -1 here at any point, he was using a super fast shortcut for FOILing a hairy set of expressions.

What he does do is use the pattern (a + b)(a - b) = (a² - b²) to quickly multiply the expressions.
Notice in a + b that both a and b are positive. Notice that in a - b, we are subtracting that positive b from a positive a. Finally notice that the result is b² subtracted from a²
So, the pattern is to take a, which is √2 and square that to get 2, THEN find the result of 1² to get 1, then subtract the results: 2 - 1 = 1
AND, for the second pair, again square a, which is √3 and square that to get 3, THEN find the result of 1² to get 1 again, and subtract that pair of results: 3 - 1 = 2
The last step was to multiply the 1 ∙ 2 for the grand finale of ` 2 `
(1 vote)
• i found the first question difficult mind breaking it down • I understand what Sal is saying in Q191 about maximising the radius (because that term is squared, but given the cylinder does't perfectly match the footprint of the box (i.e. there is waste), does that intuition necessarily follow? E.g. You could have 6 x 8 x 20 where r = 3 cylinder has greater volume than r = 4 one. • The intuition that Sal refers to is the concept, "In order to maximize volume of a cylinder, do you want to have a lengthy height, or a hefty radius?"
The volume of the cylinder is pretty simple: the radius squared times the height (times π, of course). So the radius contributes more than the height, since the radius is squared. Yes, if you get a long enough cylinder, the height is going to make a big enough difference, but `if the numbers are fairly close,` you can get a lot more volume by having a little more radius.

Let's look at the three possible side sizes for this 6 x 8 x 10 carton:
6 x 8 → the maximum possible diameter to fit within this side is 6, so the maximum radius is 3. The length of the cylinder is the remaining dimension of 10.
This volume will be 3²∙10π = 90 π
6 x 10 → the maximum possible diameter to fit within this side is still 6, so the maximum radius is still 3. The length of the cylinder is the remaining dimension of 8.
This volume will be 3²∙8π = 72 π
8 x 10 → the maximum possible diameter to fit within this side is 8, so the maximum radius is 4. The length of the cylinder is the remaining dimension of 6.
This volume will be 4²∙6π = 96 π, the winner!

Since the test-makers are testing the concept that radius squared contributes more than a linear value, they probably will not have an example like yours. An interesting question is `how much bigger does the 3rd dimension have to be in order to cause the concept to break down?`
(1 vote)
• I had a question about Problem 191, I don't know if the person who wrote this problem has ever played with those stack-able shapes as a child, but you can't put something that is 8 inches, inside something else that is also 8 inches. Therefore the answer would have to be something that is less than a radius of 4. So the answer should be radius of 3 ( if there is not a choice like 3.9) right?
(1 vote) 