Radical Expression Simplifier
Simplifies radical expressions of the form c·ⁿ√radicand by extracting perfect nth-power factors. Ideal for algebra homework, rationalising denominators, and pre-calculus problem sets.
About this calculator
A radical expression c·ⁿ√x is simplified by pulling out any factor of x that is a perfect nth power. The process: repeatedly divide the radicand by iⁿ for each prime i; each successful division moves one copy of i outside the radical while the radicand shrinks. The simplified form is (c · k)·ⁿ√r, where k is the product of all extracted bases and r is the remaining radicand containing no perfect nth-power factors. For square roots (n = 2) this means finding the largest perfect-square divisor; for cube roots (n = 3), the largest perfect-cube divisor. Negative radicands are valid only for odd indices, producing a negative result: −c·ⁿ√|x|. The numerical decimal approximation equals the exact simplified value.
How to use
Simplify 3·√72. Enter radicand = 72, index = 2, coefficient = 3. Step 1 — factor 72: 72 = 4 × 18 = 4 × 9 × 2 = 36 × 2. Step 2 — extract the perfect square: √72 = √(36 × 2) = 6√2. Step 3 — multiply by the outside coefficient: 3 × 6√2 = 18√2. The decimal value is 18 × 1.4142 ≈ 25.46. Enter those values into the calculator to confirm the result instantly.
Frequently asked questions
How do you simplify a cube root radical expression step by step?
To simplify ⁿ√x for n = 3, find the largest perfect cube that divides the radicand. For example, ³√54 = ³√(27 × 2) = 3·³√2. Factor the radicand into primes and group prime factors in triples; each complete group of three comes out as one factor in front of the radical. Any remaining primes stay inside. The calculator automates this by iterating over candidate bases and checking divisibility by their cube.
What does it mean when a radical expression cannot be simplified further?
An expression ⁿ√x is already fully simplified when the radicand contains no perfect nth-power factor greater than 1. For example, √30 = √(2 × 3 × 5) has no repeated prime factors, so it cannot be simplified. The coefficient outside the radical stays as-is. In this case the calculator returns the original expression with a decimal approximation, confirming that no simplification is possible.
Why is a negative radicand only valid for odd root indices?
Even roots of negative numbers are not real — √(−4) requires imaginary numbers because no real number squared gives a negative result. Odd roots, however, are defined for negatives: ³√(−8) = −2 because (−2)³ = −8. The sign simply carries through for odd indices. This calculator handles odd-index negative radicands by computing the root of the absolute value and prepending a minus sign to the simplified result.