Last updated on September 25th, 2020

The radical symbol **(β)** represents the square root of a number. You can multiply any two radicals that have the same indices (degrees of a root) together. If the radicals do not have the same indices, you can manipulate the equation until they do. Here is how to multiply radicals with or without coefficient.

## How to **Multiply Radicals Without Coefficients**

**Radicals need to have the same index before you multiply them.**

For Example: β(16) x β(4) = ?**Multiply the numbers under the radical signs.**

For Example: β(16) x β(4) = β(64)**Simplify radical expressions.**

β(64) = 8. 64 is a perfect square because it is the product of 8 x 8. The square root of 64 is simply 8.

## How to **Multiply Radicals with Coefficients**

For Example: Β 4β(3) x 3β(6) = 12β(Β ? )**Multiply the coefficients.**- 4 x 3 = 12

- 4 x 3 = 12
**Multiply the numbers inside the radicals.**For Example: 4β(3) x 3β(6) = 12β(3 x 6) = 12β(18)**Simplify the product.**For Example: 12β(18) = 12β(9 x 2) = 12β(3 x 3 x 2) = (12 x 3)β(2) = 36β(2)

## How to **Multiply Radicals with Different Indices**

**Find the LCM (lowest common multiple) of the indices.**Β To find the LCM of the indexes, find the smallest number that is evenly divisible by both indices. Find the LCM of the indices for the following equation:^{7}β(5) xΒ^{2}β(2) =?The indices are 5 and 2. 14 is the LCM of these two numbers because it is the smallest number that is evenly divisible by both 7 and 2. 14/7 = 2 and 14/2 = 7. To multiply the radicals, both of the indices will have to be 14.

**Write each expression with the new LCM as the index.**

Β^{14}β(5) xΒ^{14}β(2) =Β ?**Find the number that you would need to multiply each original index by finding the LCM.**– For the expression^{7}β(5), you’d need to multiply the index of 7 by 2 to get 14.

– For the expression^{2}β(2), you’d need to multiply the index of 2 by 7 to get 14.**Make this number the exponent of the number inside the radical.**Β^{2}Β –>Β^{14}β(5) =Β^{14}β(5)^{2}^{7}Β –>Β^{14}β(2) =Β^{14}β(2)^{7}**Multiply the numbers inside the radicals by their exponents.**Β Here’s how you do it:

–^{14}β(5)^{2}Β =Β^{14}β(5 x 5) =Β^{14}β25

–^{14}β(2)^{7}Β =Β^{14}β(2 x 2 x 2 x 2 x 2 x 2 x 2) =^{14}β128**Place these numbers under one radical.**Β Here’s what the result would look like:Β^{14}β(128 x 25)**Multiply them.**Β^{14}β(128 x 25) =^{14}β(3200).