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I've been calculating fractional exponents my whole academic life without really questioning why they can be calculated/defined the way they are. Let's say we have $9^\frac{1}{2}$. This equals $3$, the square root of nine. Wy this is true is that it has been defined this way because if we have $9^\frac{1}{2}*9^\frac{1}{2}$ we invoke the additive law of the exponents to get $1$ on the exponent and therefore $9$ as the result.

But why do we assume that the law of adding up exponents also works with fractional exponents? It makes intuitive sense when we have whole numbers, but why do we assume we can also add the fractional exponents like with whole number exponents?

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The short answer: if we can choose to define fractional exponents however we want, why not define them consistent with properties of integer exponents?

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