How to multiply radicals with the same index

When two radicals have the same kind of root, we can often put their product under one radical sign. Let’s see how that works for square roots, then try it with coefficients and higher roots.

Multiply what’s under the square roots

The radicand is the number or expression under the radical sign. If a≥0 and b≥0, the product of their real square roots is the square root of their product.

a·b=ab3·2=3·2=6

The rule works in both directions. For example, factoring 12 as 4·3 lets us write 12 as 4·3=23.

Multiply two identical square roots

5·5=25=5

For any nonnegative a, two copies of a multiply to a. That follows from the definition of the square root.

Keep outside coefficients outside the radical

In (42)·3, the 4 multiplies the whole product. Multiply the roots and carry the 4 along as a factor.

(42)·3=42·3=46

Apply the rule to cube roots and other matching roots

The index tells you the kind of root. A cube root has index 3, and a fourth root has index 4. With matching odd indices, the product rule works for all real radicands. With matching even indices, the radicands must each be nonnegative for the real roots to exist.

(-213)3=-21(134)4=13

In each of these examples, raising the root to its own index gives back the radicand.

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