Multiplying Radicals Calculator
Enter the coefficient, radicand, and index for each of two radical expressions and get the simplified product. The calculator handles same-index radicals using the standard product rule and different-index radicals by converting both to a common index via the least common multiple. It shows every step of the simplification so you can follow the working.
Formula
Worked example
Multiply 3*sqrt(12) by 2*sqrt(3). Both are square roots (index 2). Multiply coefficients: 3 x 2 = 6. Multiply radicands: 12 x 3 = 36. Result: 6*sqrt(36). Simplify: sqrt(36) = 6. Final answer: 6 x 6 = 36.
How to multiply radical expressions
Multiplying radicals follows two core rules depending on whether the radicals share the same index. When both radicals have the same index n, the product rule states that the nth root of a times the nth root of b equals the nth root of (a times b). After combining the radicands under one radical, you simplify by pulling out any perfect nth-power factors. For example, 3*sqrt(12) times 2*sqrt(3) gives 6*sqrt(36), and since sqrt(36) = 6, the final result is 36. When the radicals have different indices, you first convert both to a common index using the least common multiple (LCM) of the two indices, then apply the product rule normally.
Simplifying after multiplication
After the product rule gives you a single radical, simplify the radicand by factoring out perfect nth powers. For a square root (index 2), look for perfect square factors such as 4, 9, 16, or 25 inside the radicand. For a cube root (index 3), look for perfect cube factors such as 8, 27, or 64. Pull each such factor outside the radical by taking its root and multiplying it into the outer coefficient. A radicand is fully simplified when it contains no factor that is a perfect nth power (other than 1). For instance, sqrt(72) factors as sqrt(36 times 2), and since 36 is 6 squared, it simplifies to 6*sqrt(2).
Multiplying radicals with different indices
When the two radicals have unlike indices, convert each to the common index L, which is the LCM of the two indices. Rewrite the nth root of a as the Lth root of a^(L/n), and similarly for the other radical. For example, the square root of 2 times the cube root of 4 uses L = LCM(2,3) = 6. Convert: sqrt(2) = 6th-root(2^3) = 6th-root(8), and cube-root(4) = 6th-root(4^2) = 6th-root(16). Now multiply under the common radical: 6th-root(8 times 16) = 6th-root(128). Simplify: 128 = 2^7, and since 6 divides into 7 once with remainder 1, extract one factor of 2: 2 times 6th-root(2).
Multiplying coefficients with radicals
When each radical expression has a coefficient in front, multiply the coefficients separately from the radicals. Given p*nth-root(a) times q*nth-root(b), the result is (p times q) times nth-root(a times b). For instance, 5*sqrt(6) times 4*sqrt(6) = 20*sqrt(36) = 20 times 6 = 120. If the resulting radicand can be simplified further, apply the simplification rules above to get the final form. This separation of coefficients from the radical part keeps the arithmetic organized and reduces the chance of error.
Multiplying radicals: key rules
| Rule | Formula | Example |
|---|---|---|
| Product rule (same index) | n-root(a) x n-root(b) = n-root(a x b) | sqrt(3) x sqrt(12) = sqrt(36) = 6 |
| Coefficient rule | p * n-root(a) x q * n-root(b) = (pq) * n-root(ab) | 2*sqrt(5) x 3*sqrt(5) = 6*sqrt(25) = 30 |
| Different-index rule | n-root(a) x m-root(b) = L-root(a^(L/n) x b^(L/m)) | sqrt(2) x cbrt(4) = 6-root(4 x 16) = 6-root(64) |
| Perfect-power extraction | n-root(k^n x r) = k * n-root(r) | sqrt(12) = sqrt(4 x 3) = 2*sqrt(3) |
| Integer result | n-root(a^n) = a (a >= 0) | cbrt(8) = 2, sqrt(49) = 7 |
These identities apply whenever a and b are non-negative real numbers.
Frequently asked questions
Can you multiply radicals with different indices?
Yes. When the indices differ, find the least common multiple (LCM) of both indices to get a common index. Convert each radical to that common index by raising its radicand to the appropriate power, then apply the product rule. For example, to multiply the square root of 2 by the cube root of 4, use LCM(2,3) = 6 as the common index, rewrite both as sixth roots, then multiply the radicands.
Why do I need to simplify after multiplying?
After the product rule combines two radicands under one radical sign, the new radicand may contain perfect nth-power factors that were not obvious in the original expressions. Extracting those factors gives the standard simplified form, which is easier to compare and use in further calculations. Leaving a simplifiable radical such as sqrt(12) in place of 2*sqrt(3) is technically correct but not in simplest form.
What does "simplest radical form" mean?
A radical expression is in simplest form when the radicand contains no factor (other than 1) that is a perfect nth power for the given index. For square roots this means no perfect square factors inside; for cube roots, no perfect cube factors inside; and so on. The coefficient outside the radical should be a rational number, and if the index and the exponent of the radicand share a common factor, the index should be reduced.
Can radicands be decimals or fractions?
Yes, though this calculator focuses on integer radicands for clean simplification. You can enter a decimal radicand and the calculator will still compute the decimal result correctly. Simplification into a clean radical form is only shown when the radicand (after applying the common index) is a whole number, because prime factoring non-integers is not uniquely defined.
What is the product rule for radicals?
The product rule states that the nth root of (a times b) equals the nth root of a times the nth root of b, provided a and b are non-negative. Reading it in reverse, it lets you combine two radicals with the same index into one radical, which you then simplify. This rule is derived from the property of exponents: (ab)^(1/n) = a^(1/n) times b^(1/n).