Multiplying Radicals Calculator

Understanding Radical Multiplication

Multiplying radicals is a fundamental algebraic operation used throughout mathematics, physics, and engineering. A radical expression involves a root symbol ($\sqrt{}$ or $\sqrt[n]{}$), where $n$ is the index (degree of the root) and the value under the radical is called the radicand.

The golden rule for multiplying radicals with identical indices is the Product Property of Radicals:

$$\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \quad (a, b \ge 0 \text{ if } n \text{ is even})$$

When numerical coefficients are present outside the radicals, multiply outside numbers together and inside numbers together:

$$(c_1 \sqrt[n]{r_1}) \cdot (c_2 \sqrt[n]{r_2}) = (c_1 \cdot c_2) \sqrt[n]{r_1 \cdot r_2}$$

Step-by-Step Method for Multiplying Radicals

To multiply and simplify any radical expression:

1. Multiply outer coefficients: Calculate the product of all constants in front of the radical signs ($c_1 \cdot c_2$).
2. Unify the radical index: If indices match ($n_1 = n_2 = n$), keep the common root. If indices differ, determine the least common multiple $\text{LCM}(n_1, n_2)$ and convert radicands using fractional exponent rules: $\sqrt[n]{x} = \sqrt[L]{x^{L/n}}$.
3. Multiply the radicands: Multiply the numbers inside the root symbol together ($r_1 \cdot r_2$).
4. Simplify the product: Find prime factors of the combined radicand and pull out any perfect $n$-th powers ($p^n \implies p$ outside the root).

Multiplying Radicals with Different Indices

When multiplying radicals of differing root indices (for example, $\sqrt{2} \cdot \sqrt[3]{3}$), you cannot immediately multiply their inner numbers. Instead:

1. Find the Least Common Multiple (LCM) of the indices: $\text{LCM}(2, 3) = 6$.
2. Rewrite each term with index 6: $$\sqrt{2} = 2^{1/2} = 2^{3/6} = \sqrt[6]{2^3} = \sqrt[6]{8}$$ $$\sqrt[3]{3} = 3^{1/3} = 3^{2/6} = \sqrt[6]{3^2} = \sqrt[6]{9}$$
3. Multiply under the unified 6th root: $$\sqrt[6]{8} \cdot \sqrt[6]{9} = \sqrt[6]{72}$$

Multiplying Radical Binomials (FOIL Method)

When multiplying expressions with two terms, such as $(a\sqrt{x} + b\sqrt{y})(c\sqrt{z} + d\sqrt{w})$, apply the FOIL principle:

• First: $(a\sqrt{x}) \cdot (c\sqrt{z}) = ac\sqrt{xz}$
• Outer: $(a\sqrt{x}) \cdot (d\sqrt{w}) = ad\sqrt{xw}$
• Inner: $(b\sqrt{y}) \cdot (c\sqrt{z}) = bc\sqrt{yz}$
• Last: $(b\sqrt{y}) \cdot (d\sqrt{w}) = bd\sqrt{yw}$

After multiplying all four terms, simplify each individual radical and combine any like radicals (terms with the same index and radicand) and rational integer constants.

For division of roots and rationalizing denominators, explore our Dividing Radicals Calculator or use our Simplify Radical Expressions tool. You can also explore our Multiplying Polynomials Calculator for general algebraic products.