How to Multiply Radicals

Posted by
Multiply Radicals

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. 

How to Multiply Radicals Without Coefficients

  1. Radicals need to have the same index before you multiply them.
    For Example: √(16) x √(4) = ?

  2. Multiply the numbers under the radical signs.
    For Example: √(16) x √(4) = √(64)

  3. 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

  1. Multiply the coefficients.
    For Example:  4√(3) x 3√(6) = 12√( ? )
    • 4 x 3 = 12

  2. Multiply the numbers inside the radicals.
    For Example: 4√(3) x 3√(6) = 12√(3 x 6) = 12√(18)

  3. 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

  1. 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.

  2. Write each expression with the new LCM as the index.
     14√(5) x 14√(2) = ?

  3. 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.

  4. Make this number the exponent of the number inside the radical. 
    2 –> 14√(5) = 14√(5)2
    7 –> 14√(2) = 14√(2)7

  5. 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

  6. Place these numbers under one radical. Here’s what the result would look like: 14√(128 x 25)

  7. Multiply them. 14√(128 x 25) = 14√(3200).