 # 298^9 CALCULATED, it's absolutely nuts

Alright, I think I am done with mental math for a while I almost fried my brain with this one.
It took me around 1,5 hours to calculate this big boy. Just look at that number! 23 digits! I’m going to dive into my bed right now, good night! xD

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How come it says 3 in a row! on the screen when on the left it says:

Total Problems: 1
Solved Problems: 1 (100%)
Failed Problems: 0 (0%)

I don’t know exactly why that happens. I think I did some smaller powers calculations before and the site remembered those results?

It’s a great website but it needs some improvement here and there. It would be great if you could for example set a specific number of power. Right now I have to reset, reset, reset and so on until I get the right power I want.

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I actually did something so dumb at the end with this one.

Instead of multiplying 298^8, which is 62,191,368,383,776,973,056, with 300 first and then substracting (298^8) x 2 (Which I did the entire calculation ), I came up with the brilliant idea to try to do them simultaneously. To save time.

I would multiply 62 trillion, 191 billiard with 300, which gave me 18 trilliard 657,3 trillion.
Subtract 62 trillion x 2, resulting in 18 trilliard 533,3 trillion.
adding to that 368 billion x 300, answer now is 18 trilliard, 533 trillion, 410,4 billiard.
Subtracting again with 191 billiard x 2, answer is 18 trilliard 533 trillion, 28, 4 billiard.
Adding again 383 milliard x 300, answer: 18 trilliard, 533 trillion, 28 billiard, 514,9 billion.
Subtract 368 billion x 2, answer: 18 trilliard, 533 trillion, 27 billiard, 778, 9 billion.
Etc…

The constant switching made it so hard to keep track of the numbers, I regretted it immediately but I couldn’t turn back anymore. Never doing that again 2 Likes

Amazing!

I’d have gone as far as “taking it off twice” instead of x2 just because it makes it easier to keep track of “the same number” without having to double it before subtracting (twice).

You mean 298^n * 300 - 298^n * 2 = 298^{n+1} right from n=1 , so basically x times 3, add two zeroes, take off x times 2 until you’ve done that 8 times.

You could have just kept squaring instead to save time… quickly with 2^8 to illustrate:

2^2=4
4^2=16
16^2=256

That way you skip a few calculations along the way: 2, 4, 8, 16, 32, 64, 128, 256

An easy way to square 298 because it’s so close to 300 is to subtract the missing 2 from 300, so 298-2=296, multiply by 3 because your base is 100 not 300 for 296*3=888, and append the 2^2=04 for 888|04 to get 298^2=88,804

Then square that for 298\color{red}^4 and square the result of that for 298\color{red}^8 and then do x300 and subtract your previous result twice. (Last step as you’ve described above, basically.)

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That is actually much more difficult for me because i.e. squaring 298^4 is 7886150416x7886150416, a 10 digit multiplication. That requires so much precision and work, the risk of making a mistake is too high.

It is easier and much more safer for me to just go 298^n x 300 - 298^n x 2.

In the future I am probably going to try a 3 digit number to the 10th+ power. Beyond 9 and 10th power, the numbers blow up. For example 298^13.

There is no easy way to square 298 to a comfortable number close to 298^13.

298^8 is 5 powers away, 298^3^3 is basically 26463592x26463592x26463592, way too risky for me. Calculating 298^16 to then go back to 298^13 requires flawless subtraction, I am better at addition than subtraction. It’s like pulling a break, it can throw me completely off.

For lower powers that method is completely fine but beyond like 8,9 and 10th power, that method becomes more and more difficult. It’s not easy to juggle with 20, 25, 30+ digits in your mind using only natural memory, better to just stay in my comfort zone for now 1 Like

I’m curious; because of all this practice with insane powers, are you able to square 3 digit numbers really easily and quickly?

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Yes, 3 digit numbers up to 4th power aren’t really a challenge anymore.

3 digit numbers to the 2nd power should be easy for everyone else here as well.

I don’t practise these large calculations, I just try and see how it goes. They take so much time and energy to calculate, it’s not even worth it to practise them.

I have better things to do, haha x)

I can imagine Then I’m one of the few, because I still very much struggle with 3 digit squares lol. What technique do you use for them? And how fast does it take you?

I don’t use any techniques for mental calculations or memory. It takes me around 10-30 seconds or less for 3 digit squares.

Example, for 673^2 I calculate:

673x600= 403,800

403,800 + 673x70 = 450,910

450,910 + 673x3= 452,929

There are many people here on the forum, for example @Kinma , who know all kinds of techniques and methods to calculate 3 digit squares or any other squares Well, you did do 298^2 as:

298x200 + 298x90 + 298x8

That sounds like you are okay with 2-digit squares. I presume you’ve seen \color{blue}(a+b)^2=a^2+2ab+b^2 in high school? That’s all you do then…

673^2=(600+73)^2=600^2+2*600*73+73^2

…the number basically looks like this 06 | 73, where the left-hand-side (06) is a and the right-hand-side (73) is b. This approach will work up to 9,999 and all you need to know are the 2-digit squares you already know.

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Very nice!

For 3*3 maybe you could try the multiplication software on Memoriad, It offers more accurate insight.
for each set of 10, the score range by±10% most of the time.

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That is quite impressive!

If I were to learn and enhance my mental calculation skills; where and how would I go about starting?

@bjoern.gumboldt

How long does it take you to do this in squaring 3 digits.

I am not the right guy to ask such a question because I know very little about mental math techniques actually x) but I think the 2 big ones are the cross-method for multiplication and division and the mental abacus technique for addition and subtraction.

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Well, this particular approach would work up to 9,999 (4-digits), so if we’re talking only about squaring 3-digit numbers, I’d split it 67|3 instead of 06|73 so you’d get:

670^2+670*3*2+3^2

…and then it’d probably take me as long as it will take you to read the next bit…

I know the squares up to 99 by heart, so 67^2=4,489
…add two 0s since you’re doing 67{\color{red}0}^2=448,9\color{red}00
…add the right-hand-side 3^2=\color{blue}9 to it for 448,90\color{blue}9

…this doesn’t really happen sequentially, so rounding up we’re at one second thus far, I guess? Now, how long will it take me to a) figure out 3 * 2 * 670 and b) add that result on top of 448,909.

6*670=(6*{\color{red}700}){\color{red}-}(6*{\color{red}30})=4,200-180=4,020

You know 6x7 is 42 and 6x3 is 18… the rest if shifting decimal places (i.e., adding zeros)… so another second maybe?

448,909+4,020=452,929

Doesn’t look like rocket surgery either, yes 8+4 means you gotta carry to the left, the rest is adding zero from the one or the other: 9+0 | 0+2 | 9+0. Another one second maybe?

I guess in total it’s a couple of seconds… somewhere under 5 seconds for sure. I wouldn’t call the whole thing a method though… it’s more a matter of knowing how to use algebra to make arithmetic come out easier.

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do you get huge headaces while doing problems like this because I get headaces doing mental math and im not a stupid person but your really good so I was just wondering (Im trying to find a way to prevent them)

I don’t get headaches but I do get lightheaded sometimes. The intense focus starts to slip a little bit after like 1 hour.

One great tip for when you want to focus on something for a long period of time is to drink some water.

In a video about studying someone said that top students often have a bottle of water with them to help them study for hours and hours.

During these large calculations I would walk to the kitchen every now and then to drink some water 2 Likes