# Evaluating exponentiation with negative base and rational exponent

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• January 8th 2013, 07:14 AM
MathCrusader
Evaluating exponentiation with negative base and rational exponent
How come

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

is equal to

$\frac {25}4$

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!
• January 8th 2013, 07:22 AM
Plato
Re: Evaluating exponentiation with negative base and rational exponent
Quote:

Originally Posted by MathCrusader
How come

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

is equal to $\frac {25}4$

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

NOTE that cannot be $\left( \frac {-125}8 \right )^{ \frac 3 2 }$ because that the square root of a negative.
• January 9th 2013, 08:41 AM
MathCrusader
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