How to Multiply Radicals
Two radicals can be multiplied directly under a single root only when they share the same index n (both square roots, both cube roots, and so on). The product rule states ⁿ√a × ⁿ√b = ⁿ√(ab): multiply the radicands together and keep the same index. Any coefficients out front multiply separately: (c₁·ⁿ√a) × (c₂·ⁿ√b) = c₁c₂ · ⁿ√(ab). This calculator multiplies your two radical expressions, then simplifies the result by pulling out every perfect nth-power factor it can find.
Multiplying and then simplifying
Start by multiplying the coefficients: a × c. Then multiply the radicands: b × d. The raw product is (ac)·ⁿ√(bd). To simplify, factor bd into primes and group them in sets of n: every complete group of n identical prime factors comes out of the radical as one factor, and any leftover primes stay inside. For example, 3√8 × 2√18 = 6√144, and since 144 = 12², the square root simplifies exactly to 12, giving a final answer of 6 × 12 = 72 with nothing left under the radical.
Working with different indices
If two radicals do not share the same index — say a square root times a cube root — you cannot combine them with this rule directly. Convert each radical to a rational exponent (ⁿ√x = x^(1/n)), find a common denominator for the exponents, rewrite both as radicals of that common index, and then multiply. This calculator assumes both radicals already share the index you select.
Notes on negative radicands
- Even index (square root, 4th root, ...): the radicand must be zero or positive — an even root of a negative number is not a real number.
- Odd index (cube root, 5th root, ...): negative radicands are allowed and produce a real, negative result, since an odd power of a negative number stays negative.
- Rounding: avoid rounding the radicand or coefficients before multiplying — simplify the exact product first, then convert to a decimal if you need one.