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In terms of mathematical concepts, there is no difference at all. In practical terms, some rational exponents or rational number will result in rational answers while radical exponent won't. But that is hardly a significant difference.
The laws of exponents work the same with rational exponents, the difference being they use fractions not integers.
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This exact question is on a puzzle worksheet over rational exponents used by teachers. The answer to the puzzle is Nicole Oresme.
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