Rational exponent
SAT glossary · Math
Definition
x^1/n=sqrt[n]x and x^m/n=(sqrt[n]x)^m=sqrt[n]x^m — radicals and fractional exponents are the same operation in two costumes, and the entire skill is fluent translation, because the examination writes the expression in one notation and puts the answer in the other. The denominator is the root, the numerator is the power. Fractional exponents are forced by the law x^ax^b=x^a+b, not conventions.
Rational exponent in plain English
Fractional exponents and radicals (roots) are secretly the exact same mathematical operation, just written in two different 'costumes.' The rule is x^1/n=sqrt[n]x (so the DENOMINATOR of the fractional exponent tells you which root to take), and more generally x^m/n=(sqrt[n]x)^m=sqrt[n]x^m (the numerator is the power). Since the test sometimes writes a problem using radical notation and expects the answer in fractional-exponent notation (or vice versa), being fluent at instantly translating between the two forms is really the whole skill being tested here — remember it as 'denominator is the root, numerator is the power.'
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