without a calculator, how do i determine that :
32^3/5
= 8
If you want to be fancy
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! ))