Multiply radical expressions with same or different indices. Simplify √a·√b, ⁿ√a·ⁿ√b, and mixed-index radicals with step-by-step simplification.
Multiplying radicals is a key algebra skill that appears in simplification, rationalizing denominators, and solving radical equations. When two radicals share the same index, the product rule lets you combine them under a single radical: ⁿ√a · ⁿ√b = ⁿ√(ab). When the indices differ, you must first convert to a common index using rational exponents before multiplying. This calculator handles both cases. Enter the radicands and indices for two radical factors and instantly see the raw product, the simplified radical form, and the decimal approximation. The step-by-step display walks you through prime factorization, factor extraction, and index unification so you can learn the process — not just get an answer. A built-in reference table summarizes the key radical multiplication rules, and comparison bars let you visually compare the magnitudes of each factor and the product. Use the preset buttons to explore textbook-classic examples like √2·√8, ³√4·³√2, and mixed-index products without entering values manually.
Multiplying Radicals Calculator helps you solve multiplying radicals problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Coefficient (c₁), Radicand (a), Coefficient (c₂) once and immediately inspect Factor A, Factor B, Raw Product Radicand to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Same index: ⁿ√a · ⁿ√b = ⁿ√(a·b). Different indices: a^(1/m) · b^(1/n) = a^(n/mn) · b^(m/mn) = ᵐⁿ√(aⁿ · bᵐ). Simplification extracts perfect nth-power factors.
Result: Factor A shown by the calculator
Using the preset "Same Index", the calculator evaluates the multiplying radicals setup, applies the selected algebra rules, and reports Factor A with supporting checks so you can verify each transformation.
This calculator takes Coefficient (c₁), Radicand (a), Coefficient (c₂), Radicand (b) and applies the relevant multiplying radicals relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Factor A, Factor B, Raw Product Radicand, Simplified Form to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Yes. Convert each radical to a rational exponent, find a common denominator for the exponents, then combine under a single radical with the new common index.
ⁿ√a · ⁿ√b = ⁿ√(a·b), provided both radicands are non-negative when n is even. Use this as a practical reminder before finalizing the result.
Factor the radicand into a perfect nth power times a remaining factor, then extract the perfect power from under the radical. Keep this note short and outcome-focused for reuse.
Negative radicands are allowed only for odd indices. For even indices (square roots, 4th roots, etc.), negative radicands produce complex numbers.
No. Radical multiplication is commutative: √a·√b = √b·√a.
A surd is an irrational root expression that cannot be simplified to a rational number, such as √2 or ³√5. Apply this check where your workflow is most sensitive.