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Quotient to a Power Property for Exponents
The Quotient to a Power Property for Exponents provides a shortcut for raising a fraction to a power. If and are real numbers, , and is a whole number, then:
In words: to raise a fraction to a power, raise both the numerator and the denominator to that power separately. The property follows from the definition of exponents applied to fraction multiplication. Consider : by definition, this means . Multiplying the numerators together and the denominators together gives . Each part of the fraction is raised to the exponent individually.
A numerical check confirms the rule: . The left side gives , and the right side gives ✓.
This property is the division counterpart to the Product to a Power Property, which distributes an exponent across factors in a product.
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