Best way of working this out without a calculator

mufc802

mufc802

mufc802

Soldato
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(343/512)^-2/3 (343/512)^(-2/3) (sorry forgot brackets)

I firstly flipped it over to give a positive power making:
(512/343)^2/3 (512/343)^(2/3)

Now I did:
3root512/343^2 3root(512/343)^2

3root262144/117649

But what is the way of working that out without a calculator?
 
Last edited:
Abacus_2.gif
 
well, you should do the root first, 512 is obvously a power of 2
2^9 = 512 so
(2^3)^3 = 512
so the cube root of 512 is 8 which square to 64

don't know a simple way of doing the cube root or 343, but I did recognise it as being a cubed number, tried 7, it worked.

so the answer is 64/49


much easier to do wolfram alpha though
 
(343/512)^(-2/3)

1 / (343/512)^(2/3)

1 / (cbrt(343/512))^2

Cube root of 343 is 7, cube root of 512 is 8, so,

1 / (7/8)^2

1 / (49/64)

64/49

edit.....and beaten...twice :p
 
mufc802 said:
OK but how did you work out the cube root of 512 and 343 in your head? If the number was much greater how do you do it without a calculator?
Click to expand...

Same way Judgeneo said really. There's no particularly 'easy' way of doing the first 10 cube numbers, you just have to learn them. 512 happens to be a bit easier as it's a factor of 2, however the numbers like 343 are a bit of a pain.

http://www.1729.com/blog/CubeRoots.html
 
You don't really need to learn them if you don't want to. It's pretty easy to see roughly what numbers might work and then trial and error from there.
 
Decompose your numbers into prime factors, then consider the answer

512 = 2^9
343 = 7^3

You can then apply the rules of exponents to simplify to 2^6/7^2 = 64/49 which is going to be just a little less than 4/3
 
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