55.5 Rational Exponent Laws

Rational Exponent Laws extend exponent rules to fractions, simplifying algebraic expressions with rational powers.

Rational Exponent Laws are the extensions of the standard integer exponent rules, product of powers, quotient of powers, power of a power, power of a product, and power of a quotient, to exponents that are fractions rather than whole numbers, applied by adding, subtracting, or multiplying the fractional exponents using ordinary fraction arithmetic wherever the corresponding integer law would call for adding, subtracting, or multiplying whole-number exponents. These laws hold for rational exponents in exactly the same form as for integer exponents, with the only additional consideration being the need to find a common denominator before adding or subtracting two exponents that are not already expressed over the same denominator.

Because these laws are direct extensions of already-familiar integer exponent rules, applying them to rational exponents requires no new algebraic principle beyond comfortable fraction arithmetic and, where required, converting exponents to a shared denominator first.

Combining Powers of the Same Base

Common-Base Rational Product

When multiplying two powers of the same base, their rational exponents are added together, following the identical structure of the integer product-of-powers rule.

a^{\frac{1}{2}} \cdot a^{\frac{1}{3}} = a^{\frac{1}{2} + \frac{1}{3}}

Common-Base Rational Quotient

When dividing two powers of the same base, their rational exponents are subtracted, following the identical structure of the integer quotient-of-powers rule.

\frac{a^{\frac{2}{3}}}{a^{\frac{1}{3}}} = a^{\frac{2}{3} - \frac{1}{3}} = a

Fractional Exponent Common Denominator

Before two rational exponents can be added or subtracted in either of the rules above, they are first rewritten with a common denominator, exactly as any two fractions would be, ensuring the addition or subtraction can be carried out correctly.

\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Powers of Powers, Products, and Quotients

Rational Power of a Power

When a base already raised to a rational exponent is itself raised to another rational exponent, the two exponents are multiplied together, following the identical structure of the integer power-of-a-power rule.

{(a^{\frac{1}{2}})}^{\frac{1}{3}} = a^{\frac{1}{2} \cdot \frac{1}{3}} = a^{\frac{1}{6}}

Rational Power of a Product

When a product of two bases is raised to a rational exponent, that exponent distributes across both bases individually, following the identical structure of the integer power-of-a-product rule.

{(a b)}^{\frac{1}{2}} = a^{\frac{1}{2}} \cdot b^{\frac{1}{2}}

Rational Power of a Quotient

When a quotient of two bases is raised to a rational exponent, that exponent distributes across both the numerator and denominator individually, following the identical structure of the integer power-of-a-quotient rule.

{(\frac{a}{b})}^{\frac{1}{2}} = \frac{a^{\frac{1}{2}}}{b^{\frac{1}{2}}}

Handling Negative Exponents Within These Laws

Negative Rational Exponent Conversion

When a negative rational exponent appears within an expression being simplified using these laws, it is first converted to a positive rational exponent applied to the reciprocal of the base, exactly as with any negative exponent, before the relevant law is applied.

a^{- \frac{1}{2}} = {(\frac{1}{a})}^{\frac{1}{2}}

Applying the Laws Correctly

Domain-Safe Exponent Law Use

Before applying any of these laws, the base is checked against the rational power domain condition relevant to the exponents involved, since these laws assume the underlying expressions are already defined over the real numbers, and applying them to an expression outside that domain can produce a nonsensical or contradictory result.