Radical form
Also known as: simplified radical form, radical notation
Radical form is a way of writing roots using the radical symbol (√) instead of fractional exponents. For example, x^(1/2) in exponent form is √x in radical form.
Radical form expresses a root with the radical symbol rather than a fractional exponent. The two notations are interchangeable: x^(m/n) equals the nth root of x^m. The denominator of the fractional exponent becomes the index of the radical (the small number tucked into the radical sign), and the numerator stays as the power. So x^(1/2) = √x, x^(1/3) = ∛x, and x^(2/3) = ∛(x²).
Converting between the forms is a mechanical skill worth drilling. Going from radical to exponent form: ∜(x³) becomes x^(3/4) — index in the denominator, power in the numerator. Going the other way: 8^(2/3) is the cube root of 8², which is the cube root of 64, or 4. (Computing the root first — cube root of 8 is 2, then 2² = 4 — usually keeps the numbers smaller.)
An answer in simplified radical form follows a few conventions: no perfect-power factors left under the radical (√12 simplifies to 2√3, since 12 = 4 × 3), no fractions under the radical, and no radicals in a denominator — those are cleared by rationalizing, so 1/√2 becomes √2/2. Test answer choices are almost always written this way, so an unsimplified result may not match any option even when it's correct.
Radicals and rational exponents appear throughout the ACT and SAT math sections. Both exams ask you to convert between radical and exponent form, simplify radical expressions, and combine them with the exponent rules covered in Achievable's algebra chapters.
Key takeaways
- Radical form writes roots with the √ symbol; fractional exponents express the same thing: x^(m/n) = nth root of x^m.
- The denominator of a fractional exponent becomes the radical's index; the numerator stays as the power.
- Simplified radical form leaves no perfect-power factors under the radical and no radicals in denominators.
- √12 = 2√3, and 1/√2 rationalizes to √2/2.