Why is 5-⅔ (5 raised to the power of -2/3) equal to 1/5⅔ (1 divided 5 raised to the power of 2/3)? I mean to say how does the negative sign of the exponent ⅔ change to positive when 5-⅔ (5 raised to the power of -2/3) is inversed?
Thanks,
Ron
Because it is. You can explain it using this formula :
$\displaystyle \frac{a^m}{a^n} = a^{m - n}$
In our case, $\displaystyle m = 0$ because $\displaystyle a^0 = 1$, and thus it follows that :
$\displaystyle \frac{a^0}{a^n} = a^{0 - n}$
Which is equivalent to :
$\displaystyle \frac{1}{a^n} = a^{-n}$
Now, you are probably going to ask us why $\displaystyle \frac{a^m}{a^n} = a^{m - n}$. To this I will answer : "Why does 1 + 0 = 1 ?".
I wouldn't say "It's just obvious". When you have $\displaystyle 5^2$, it means $\displaystyle 1*5*5$. When the exponent, which means the number of 5s you multiply the 1 by is negative, you start dividing. Therefore, $\displaystyle 5^{-1}=\frac {1} {5}$