Rational Exponents and Radical Form

Think about what an exponent of 1/2 has to mean if the rule for multiplying powers still has to work. That rule says x^a times x^a equals x^2a. So x^1/2 times x^1/2 must equal x^{2 times 1/2}, which is x¹, or just x. But the number that multiplies by itself to give x is exactly the square root of x. So x^1/2 has to mean √x — there's no other value it could be and still obey the rules of exponents.

That same logic stretches to any rational exponent. For x^p/q, the denominator q tells you which root to take — the qth root — and the numerator p tells you what power to raise to, in either order. So x^p/q equals the qth root of x^p, which also equals (the qth root of x) raised to the p. Written the other way: a root sits on the outside as a fractional exponent with that root's index on the bottom. The cube root of x⁵, for instance, is x^5/3: index 3 on the bottom, power 5 on top.

This is the bridge between radical expressions and power expressions, and it matters because every rule you already know for exponents — multiplying, dividing, raising a power to a power — now works on roots too, as long as you first rewrite the root as a fractional exponent. That's exactly the move that simplifies expressions with nested radicals, and it's also what lets you solve equations where the unknown sits under a root: rewrite the root as a fractional power, then undo it by raising both sides to the reciprocal exponent.