Multiplication of Radicals

The product of two radicals with the same index is itself a radical. Its radicand is obtained by multiplying the radicands, while the index remains unchanged. $$ \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} $$ Here, a and b are non-negative real numbers, and n is a nonzero natural number.

Radicals can be multiplied directly only when they share the same index.

If the indices are different, each radical must first be rewritten using the least common index.

A Practical Example

Example 1

Consider the product of two radicals

$$ \sqrt[3]{4} \cdot \sqrt[3]{5} $$

Because the radicals have the same index, multiply the radicands and retain the index

$$ \sqrt[3]{4} \cdot \sqrt[3]{5} = \sqrt[3]{4 \cdot 5} = \sqrt[3]{20} $$

Example 2

Now consider the product

$$ \sqrt[3]{4} \cdot \sqrt{5} $$

In this case, the radicals do not share the same index.

First, rewrite them using the least common index, obtained by computing the least common multiple of the indices.

$$ \mathrm{lcm}(3,2) = 6 $$

Next, apply the invariance property of radicals to express both radicals with index 6

$$ \sqrt[3 \cdot 2]{4^2} \cdot \sqrt[2 \cdot 3]{5^3} $$

$$ \sqrt[6]{4^2} \cdot \sqrt[6]{5^3} $$

Finally, multiply the radicals, which now have the same index

$$ \sqrt[6]{4^2 \cdot 5^3} $$

Proof of the Rule

We now prove the multiplication rule for radicals

$$ \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} $$

Raise both sides of the equation to the nth power

$$ ( \sqrt[n]{a} \cdot \sqrt[n]{b} )^n = ( \sqrt[n]{a \cdot b} )^n $$

Apply the laws of exponents.

Simplify the radical index n with the exponent n on the right-hand side.

$$ ( \sqrt[n]{a} \cdot \sqrt[n]{b} )^n = ( \sqrt[{ \not{n} }]{a \cdot b} )^{ \not{n} } $$

$$ ( \sqrt[n]{a} \cdot \sqrt[n]{b} )^n = a \cdot b $$

Apply the exponent rule for products with equal exponents (a \cdot b)^x = a^x \cdot b^x to the left-hand side

$$ ( \sqrt[n]{a} )^n \cdot ( \sqrt[n]{b} )^n = a \cdot b $$

Now simplify each radical index with its exponent.

$$ ( \sqrt[{ \not{n} }]{a} )^{ \not{n} } \cdot ( \sqrt[{ \not{n} }]{b} )^{ \not{n} } = a \cdot b $$

$$ a \cdot b = a \cdot b $$

The equality holds, which establishes the validity of the rule.

And so on.