Multiplying Radicals Calculator (multiply & simplify square, cube and n-th roots)

Understanding the multiplying radicals calculator — concept and principle

This calculator multiplies two radicals with the same index, such as two square roots or two cube roots, and writes the product in simplest radical form with a decimal check. Use it for algebra homework, to check a simplification by hand, or to see each step of the product rule for roots.

How it is calculated

a·ⁿ√x · c·ⁿ√y = (a·c)·ⁿ√(xy)

• n is the index (2 = square root, 3 = cube root, up to 10).
• a and c are whole-number coefficients in front of the roots; x and y are positive whole-number radicands.
• After multiplying, the new radicand xy is split into a perfect n-th power kⁿ times a leftover m, and k moves outside: ⁿ√(kⁿ·m) = k·ⁿ√m.
• The decimal value is a·c·(xy)^(1/n).

The principle

A root is a fractional power: ⁿ√x = x^(1/n). Powers with the same exponent multiply by multiplying the bases, so x^(1/n)·y^(1/n) = (xy)^(1/n). That is the product property of roots, and it only works when both radicals share the same index. Simplifying uses the same rule backwards: if xy contains a factor that is a perfect n-th power, its root is a whole number and can be written in front. The calculator finds the largest such factor by prime factorisation: each prime's exponent is divided by n, the quotient goes outside and the remainder stays inside.

Worked example

Default inputs: n = 2, first radical 3√6, second radical 2√12.

1. Multiply the coefficients: 3 × 2 = 6.
2. Multiply the radicands: 6 × 12 = 72, giving 6√72.
3. Factor 72 = 36 × 2, and 36 = 6² is the largest perfect square factor.
4. √72 = √36 · √2 = 6√2, so 6√72 = 6 × 6√2 = 36√2.
5. Decimal check: 6 × √72 ≈ 50.91168825.

How to read the result

The simplified product is the exact answer and is what teachers usually expect. A result with no radical left (for example √8 · √2 = 4) means the product of the radicands was a perfect power. If the line "take out perfect powers" says there is nothing to take out, the radicand was already square-free (or cube-free for n = 3). Use the decimal value only to check the size of the answer or for measurements.

Common mistakes

• Multiplying a coefficient by a radicand, e.g. writing 3√6 · 2√12 as √(3·6·2·12). Coefficients multiply with coefficients only.
• Multiplying radicals with different indexes directly. Convert both to rational exponents first.
• Stopping at 6√72 without simplifying. Check for any perfect-power factor.
• Taking out a factor that is not the largest one, e.g. √72 = 2√18, and forgetting to simplify again.

Frequently asked questions

Can I multiply a square root by a cube root?

Not with the product rule directly. Write them as powers, for example √2 · ∛2 = 2^(1/2) · 2^(1/3) = 2^(5/6) = ⁶√32.

Why is √5 · √5 equal to 5?

Because √5 · √5 = √25, and 25 is a perfect square, so the root comes out completely.

Is 36√2 the same as √2592?

Yes. 36² × 2 = 2,592, so both are equal, but 36√2 is the simplest form.

Does the order of the radicals matter?

No. Multiplication is commutative, so swapping the two radicals gives the same product.

Sources: OpenStax, Intermediate Algebra 2e (2020), §8.4 Add, Subtract, and Multiply Radical Expressions