Evaluate:
9^(-3/2)
Got a mental blank, please guys help me quickly!
How is it 1/27? Please show complete working out.
Negative Exponent Intuition
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For any number $\displaystyle a\ne 0$, the number $\displaystyle a^{-1}$ is defined as such a number that $\displaystyle a\cdot{a^{-1}} = a^{-1}\cdot{a} = 1.$ If we for example take $\displaystyle 3$, then the number $\displaystyle 3^{-1}$ such that $\displaystyle 3\cdot{3^{-1}} = 3^{-1}\cdot{3} = 1 $ is clearly $\displaystyle \dfrac{1}{3}$, as $\displaystyle 3\cdot\left(\dfrac{1}{3}\right) = \left(\dfrac{1}{3}\right)\cdot{3} = 1.$ It can be easily seen that $\displaystyle a^{-1}$ is always $\displaystyle \dfrac{1}{a}$ (with the exception of $\displaystyle a = 0$, of course). So, for your case, we have $\displaystyle 9^{-\frac{3}{2}} = \dfrac{1}{9^\frac{3}{2}}$ Remember that $\displaystyle X^{\frac{Y}{Z}} = (X^{\frac{1}{Z}})^Y$, so you have $\displaystyle \dfrac{1}{9^\frac{3}{2}} = \dfrac{1}{(9^{\frac{1}{2}})^3}.$ Since $\displaystyle 9^{\frac{1}{2}} = 3$, we get $\displaystyle \dfrac{1}{3^3},$ which is of course $\displaystyle 27$.