Multiplying radicals is just matter of multiplying the numbers inside the radical signs, and putting the product inside a radical sign. Since the product of $8$ and $18$ is $144$, $\sqrt{8}.\sqrt{18}$ = $\sqrt{144}$, and since $12$ is the positive square root of 144, $\sqrt{144} = 12$.

To multiply radical expressions, one can use the product rule for radicals, which is given as

$\sqrt[n]{uv}$ = $\sqrt[n]{u}\sqrt[n]{v}$

In this section we will learn more about multiplying radicals.

Let $u$ and $v$ be real numbers, variables or algebraic expressions. If the nth roots of $u$ and $v$ are real, the following rule is true.

$\sqrt[n]{uv}$ = $\sqrt[n]{u}\sqrt[n]{v}$


$\sqrt[n]{uv}$ = $(uv)^{\frac{1}{n}}$

= $(u)^{(\frac{1}{n})}$ $(v)^{(\frac{1}{m})}$

= $\sqrt[n]{u}\sqrt[n]{v}$
Radicals with variables is in the form of $\sqrt{x}$. The variables inside the radical sign may be in any form. If the variables inside the radical sign involve division operation, the variable must not be equal to zero.

Simplifying radicals with variables is the process of simplifying a given radical expression as simple as possible.

Solved Example

Question: Simplify the following radical.
$\sqrt{49 x^{3}}$

$\sqrt{49 x^{3}}$ = $\sqrt{49.x.x.x}$

= $\sqrt{7.7.x.x.x}$ 

= $7x\sqrt{x}$


Sometimes a different way of expressing radicals makes a solution easier to come by. One can rewrite every radical as an exponent by using following property

$x^{\frac{m}{n}}$ = $\sqrt[n]{x^{m}}$ = $(\sqrt[n]{x})^{m}$

Example: $\sqrt[3]{8^{2}}$ = $(\sqrt[3]{8})^{2}$ = $8^{\frac{2}{3}}$
To multiply radical expression having only one term, we multiply the coefficients and multiply the radicals separately and then simplify the result, when possible.

Solved Example

Question: Simplify $3\sqrt{6}.4\sqrt{3}$
= $3.(4) \sqrt{6}\sqrt{3}$

= $12 \sqrt{18}$

= $12 \sqrt{9 \times 2}$

= $12.(3)\sqrt{2}$

= $36 \sqrt{2}$


The following are the examples of multiplying radicals

Solved Examples

Question 1: Find the product and simplify. 

$\sqrt[3]{5}.\sqrt[3]{16}$ = $\sqrt[3]{5 \times 16}$

= $\sqrt[3]{80}$

= $\sqrt[3]{8 \times 10}$

= 2$\sqrt[3]{10}$


Question 2: Find each product and simplify.
$\sqrt{3}.(2 + \sqrt{5})$

$\sqrt{3}.(2 + \sqrt{5})$ = $2 \sqrt{3}+ \sqrt{3}\sqrt{5}$

= $2 \sqrt{3}+ \sqrt{15}$