MHB What Are Some Tricks To Calculate n^3?

[susanto3311]

hi all..

how to easy calculate this problem :

17^3 = ...
13^3 =...
31^3= ...

do you have simple method to figure it out?

 

[Nono713]

Well, you just have to multiply 17 with itself twice, that is, 17^3 = 17 * 17 * 17... then you can use whatever method you want to do it "easily"/quickly. For instance if I did not have access to a calculator what I would do is start with one multiplication like this:

17 * 17 = (20 - 3) * (20 - 3) = 20 * 20 - 20 * 3 - 20 * 3 + 3 * 3 = 400 - 60 - 60 + 9 = 289

Then do the second multiplication as:

17 * 17 * 17 = 289 * 17 = (300 - 11) * (20 - 3) = 300 * 20 - 300 * 3 - 11 * 20 + 11 * 3 = 6000 - 900 - 220 + 33 = 4913

And that's it. Same goes for 13 or 31 ( the trick here is working out what you should round each number to make sure you're actually making the problem simpler and not harder, multiples of 10 or 100 are usually a good bet). If you have a calculator you can just punch it in though. But really it's just multiplication.

 

[MarkFL]

You can also use the binomial expansion:

$$(a+b)^3=a^3+3a^2b+4ab^2+b^3$$

So, for example, you would find:

$$31^3=(30+1)^3=30^3+3\cdot30^2\cdot1+3\cdot30\cdot1^2+1^3=27000+2700+90+1=29791$$

Or you could use a difference of cubes formula:

$$a^3-b^3=(a-b)\left(a^2+ab+b^2\right)$$

So, for example, we could write:

$$31^3=\left(31^2-30^3\right)+30^3=(31-30)\left(31^2+31\cdot30+30^2\right)+30^3=961+930+900+27000=29791$$