without a calculator, how do i determine that :

32^3/5

= 8

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- Apr 1st 2010, 06:26 PMdymand68evaluation of exponential expression
without a calculator, how do i determine that :

32^3/5

= 8 - Apr 1st 2010, 06:33 PMmr fantastic
- Apr 1st 2010, 06:34 PMtonio
- Apr 1st 2010, 06:54 PMdymand68
the problem is:

evaluate:

32^-3/5

=1/32^3/5

=???

1/8 ?

thus my original question: without a calculator, how do i Know that

32^3/5

= 8

...okay, find the fifth root of 32 and cube it, gotcha - Apr 1st 2010, 09:02 PMintegral
If you want to be fancy (Giggle)

This does not require a calculator at all!

$\displaystyle 32^{3/5}=n$

(take the log of both sides)

$\displaystyle \textrm{log}_{32}32^{\frac{3}{5}}=\textrm{log}_{32 }8$

$\displaystyle \textrm{log}_a(a)^x=x\therefore$

$\displaystyle \frac{3}{5}=\textrm{log}_{32}8$

log base change formula. (changes and base to two base 10 logs):

$\displaystyle \textrm{log}_ab=\frac{\textrm{log}(b)}{\textrm{log }(a)} \therefore$

$\displaystyle \frac{3}{5}=\frac{\textrm{log}8}{\textrm{log}32}$

$\displaystyle \textrm{log}(a)^x=x\textrm{log}a$

and

$\displaystyle \textrm{log}(ab)=\textrm{log}(a)+\textrm{log}(b)$

$\displaystyle \frac{3}{5}=\frac{3\textrm{log}2}{\textrm{log}4+\t extrm{log}8}$

$\displaystyle \frac{3}{5}=\frac{3\textrm{log}2}{2\textrm{log}2+3 \textrm{log}2}$

$\displaystyle \frac{3}{5}=\frac{3\textrm{log}2}{5\textrm{log}2}$

$\displaystyle \frac{3}{5}=\frac{3}{5}$

$\displaystyle \therefore 32^{3/5}=8$

((This is my 100th post! Whooo! :D ))