Evaluating exponentiation with negative base and rational exponent

How come

$\displaystyle \left( \frac {-125}8 \right )^{ \frac 23 }$

is equal to

$\displaystyle \frac {25}4 $

I know that it can be shown using several exponent laws but I thought those were only valid for a rational exponent as long as the base is positive. I have been told that negative bases rased to rational exponents is undefined. Somebody please help me sort this conundrum out!

Re: Evaluating exponentiation with negative base and rational exponent

Quote:

Originally Posted by

**MathCrusader** How come

$\displaystyle \left( \frac {-125}8 \right )^{ \frac 23 }$

is equal to $\displaystyle \frac {25}4 $

That really is the cube-root of a quantity squared.

NOTE that cannot be $\displaystyle \left( \frac {-125}8 \right )^{ \frac 3 2 }$ because that the square root of a negative.

Re: Evaluating exponentiation with negative base and rational exponent

Alright thanks. I suppose if the base is negative and the denominator of the exponent is even is what causes problems. Thx again