Olympiad Mathematics Trick | Pascal Triangle Application

[00:00:00]Hi, everyone.

[00:00:02]Can you simplify this without using calculator?

[00:00:07]4 + square root of 2 raised to the power of 3 - 3 + the square root of 2 raised to the power of 3.

[00:00:21]Okay, um for those of you that do not have um the idea of what Pascal's triangle is, let's apply Pascal's triangle to work this.

[00:00:35]I believe it's going to be faster and easier for us.

[00:00:40]Pascal's triangle is like this.

[00:00:43]We have one at the peak.

[00:00:47]We have one one.

[00:00:49]We have one.

[00:00:51]We have two.

[00:00:53]We have three.

[00:00:55]Sorry, we have one, right? Then we have one out here.

[00:01:01]1 + 2 is 3 here. 2 + 1 is 3. And then there's one, always one at the edge, right?

[00:01:10]So, this is um Pascal's triangle. The at this point, you have power of zero, power of one, power of two, and we have power of three.

[00:01:25]So, this means that um now that we have power of three, we are going to use this one. This will be the coefficient if we expand what we have there.

[00:01:35]Now, watch the way I will expand this.

[00:01:39]You know, we got 4 + the square root of 2 to the power of 3.

[00:01:47]Now, the expansion of this using Pascal's triangle, the coefficient of the first um term is going to be one, right? So, this one will now multiply.

[00:01:59]We have four to the power of three.

[00:02:03]You know, the highest power should be three because we are working with um everything to the power of three.

[00:02:10]So, we have four to the power of three multiplied by the second term here, which is root two.

[00:02:16]Root two will now have power of zero.

[00:02:20]So, that 3 + 0 will give us this power here, which is um three.

[00:02:26]So, we close this. Then, we go to the second term. The second term will now have coefficient of three. So, you pick your three.

[00:02:34]Then, you open brackets.

[00:02:36]Right? We open the brackets.

[00:02:39]And we are going to have um the same four.

[00:02:43]This time around, we'll reduce what take one from the power here. It's going to be two.

[00:02:48]And that means that this root two will have power of one.

[00:02:54]Okay? As the first term is increasing or decreasing, rather, the second term will be increasing its power.

[00:03:02]So, we close this.

[00:03:04]Right? Let me continue from here. We have plus.

[00:03:09]Now, the third term is going to have the coefficient of three.

[00:03:14]Then, we have four again to the power of one because here we have four to power two. We're going to have power of one, then multiplied by the square root of two. But this time around, it will have power of two.

[00:03:28]Remember, 1 + 2 should always give us the three that we have over here.

[00:03:33]Okay?

[00:03:34]Now, we have plus.

[00:03:36]We go over to the last term, which is one. The coefficient is going to be one. And then, four to the power of From four to power one, you get four to the power of zero multiplied by the square root of two to the power of three.

[00:03:54]Because it will increase from two to three.

[00:03:57]And zero plus three will still give us the three that we are supposed to have.

[00:04:02]So from here, what do we have?

[00:04:05]Let's work on this very quickly.

[00:04:08]We have 4 ^ 3 is 64 and √2 to the power of zero is one. 1 * 64 is still 64.

[00:04:18]So here, we're going to have um 64.

[00:04:23]Then we go plus. We're going to the other term.

[00:04:26]Um here we have 4 squared. 4 squared is 16. 16 * 3 is 48. So here we have 48.

[00:04:33]And then we have √2 from here. Now let's go here.

[00:04:38]We have um this can take this out. So we have 4 * 2, which is 8. 8 * 3 is 24. So here we have 24.

[00:04:49]Okay? Then from this part, 4 to the power of zero is one. 1 * 1 is one. So we have √ 2 to the power of three. √2 to the power of three is √2 * √2 * √2 and that will give us that will give us um two √3.

[00:05:10]Because √2 * √2 is √4, which is the same thing as two. Then multiply by the remaining √3, we have √3. Sorry, it's going to be √2.

[00:05:20]So we have 2 √2.

[00:05:22]And um I believe we can simplify this further.

[00:05:26]So we have um 20 64 + 24, that will give us 88.

[00:05:33]Right? Then we have plus 48 √2 + 2 √2, that will give us 50 √2.

[00:05:43]So we have 50 √2.

[00:05:45]Okay, so this is what we have from the first time. Let's go back to the top.

[00:05:54]Okay, so this is what we have in place of this right now. So, take notes.

[00:05:59]Okay, so we're going to go back to the second um the second term in the bracket.

[00:06:07]Okay, so we have dealt with this one.

[00:06:09]We're going over to this part right now.

[00:06:12]And we are still going to use um the Pascal triangle.

[00:06:16]So, we have three plus root two all to the power of three.

[00:06:22]And like we said, the first coefficient is what? One.

[00:06:26]Then it's going to multiply three. This time around is three, not four. Three to the power of three.

[00:06:33]Multiply by root two to the power of zero.

[00:06:37]Then plus the second coefficient is three.

[00:06:41]And it's going to multiply three to the power of two.

[00:06:45]From three to two, right? Then multiply by root two to the power of what? One.

[00:06:52]Okay.

[00:06:54]Remember, as this one is increasing this one is decreasing, and this is going to increase. That's how it is.

[00:07:02]So, we have um plus Okay, plus we have um the third term, which is three. The coefficient is going to be three. Then we have three to the power of one multiplied by the square root of two to the power of two.

[00:07:20]One plus one will always give us the three here.

[00:07:23]Then plus the coefficient, the last one is one.

[00:07:27]So, you pick one.

[00:07:29]Then we're going to have three to the power of from one to zero. Then multiply by the square root of two to the power of three. So, that this will increase from power two to the power of three.

[00:07:44]Now, let's work it very quickly.

[00:07:48]We have um two to the power of root two to the power of zero is one. The one times this is um 27. So, we'll write 27 here.

[00:08:00]27 then plus we go to the second term.

[00:08:04]Here we have um three squared nine. Nine times three is 27. So, here we have 27 root two.

[00:08:13]We're coming down here plus Now, this one will take this out like we did in the first um part. Now, two times three, that is um six. Six times three is 18.

[00:08:27]So, here we are going to have 18.

[00:08:31]plus From this part, what do we do?

[00:08:34]Three to the power of zero is one. So, what we're going to have here is two root three, which is the same thing as two root two like we had before.

[00:08:45]two root two So, from this part, we can add this and this, right? So, the addition is going to give us um this plus this will give us 45, right?

[00:08:59]I mean, 27 plus 18 is 45. Then plus this plus this will give us 29.

[00:09:06]And we have what?

[00:09:08]root two. 29 root two. So, this is what we have from the second part.

[00:09:13]Now, let let me go to um the top and fix them in their places.

[00:09:20]Okay, so this is what we we just um worked on. And from here, we got 88 plus 50 root two then minus what we have here is 20 um 45, yes.

[00:09:38]is 45 plus 29 root two.

[00:09:46]Okay, so we can easily work on this by removing the bracket.

[00:09:50]We have 88 plus 50 We have root two minus 45. Then we have minus 29 and we have root two.

[00:10:01]So, what do we do? This one minus this one is going to give us 43.

[00:10:08]Then 50 root two minus 29 root two is going to give us 21 root two.

[00:10:17]So, this is what we have after simplifying.

[00:10:21]Thank you. I believe you understood.