Last updated on September 25th, 2020

The radical symbol **(√)** 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.

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

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

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

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

√(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: 4√(3) x 3√(6) = 12√( ? )**Multiply the coefficients.**- 4 x 3 = 12

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

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

**Find the LCM (lowest common multiple) of the indices.**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}√(5) x^{2}√(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.**

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

– For the expression^{2}√(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.**^{2}–>^{14}√(5) =^{14}√(5)^{2}^{7}–>^{14}√(2) =^{14}√(2)^{7}**Multiply the numbers inside the radicals by their exponents.**Here’s how you do it:

–^{14}√(5)^{2}=^{14}√(5 x 5) =^{14}√25

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