hi all
i got some h/w for negative powers teatcher said :
X^-a = 1/X^a
why is that?
thanks.....
Raising a number to -1 will produce its reciprocal: $\displaystyle {\color{red}x}^{-1} = \frac{1}{{\color{red}x}}$ (this is just how it is defined)
So: $\displaystyle \underbrace{x^{-a} = \left({\color{red}x^{a}}\right)^{-1}}_{\text{Since: } \displaystyle a^{bc} = \left(a^{b}\right)^{c} } \! = \frac{1}{{\color{red}x^{a}}}$
Because, it is defined to be
$\displaystyle x^{a}x^{b}=x^{a+b},$
letting $\displaystyle b=-a$, we get
$\displaystyle x^{a}x^{-a}=x^{a-a}=x^{0}=1,$
which indicates
$\displaystyle x^{a}=\frac{1}{x^{-a}}\text{ or equivalently }x^{-a}=\frac{1}{x^{a}}$
privided that $\displaystyle x\neq0$.