There are several different situations in which we may have to multiply with square roots. There may be a mix of numbers and variables, and there may also be coefficients on our square roots. We will approach this topic by examining several examples:

Example 1:


We first must simplify the square roots if possible. Both of these simplify:


Now, keep coefficients together and bases together when multiplying.


Since our base is not a perfect square and cannot be simplified further, we are done.

Example 2:


Like before, we simplify the square roots separately first. Note that in this problem we have variables, but we will simply leave them "tacked on" on the ends of the bases.


We are not yet done with this problem. Look at our base. The y and z cannot be altered, however we know what √9 is. We must bring a 3 out.


Notice, when we brought the 3 out, we had to multiply it with our existing coefficient of 20. We only took the square root of the 9, not the y or z, so they must stay in the radical.

Practice Problems

1. Simplify √18 x √180.

2. Simplify √28y x √44z.

3. Destiny incorrectly worked a math problem. Based on her work below, what did she do wrong? What is the correct answer?

√24 x √8 = √( 4 x 6 ) x √( 4 x 2 ) = 2√6 x 2√2 = (2 x 2)√(6 x 2) = 4√12

Answers

1. √18 = √(9 x 2 ) = 3√2

√180 = √(36 x 5) = 6√5

3√2 x 6√5 = (3 x 6)√(2 x 5) = 18√10

2. √28y = √(4 x 7y) = 2√7y

√44z = √(4 x 11z) = 2√11z

2√7y x 2√11z = (2 x 2)√(7y x 11z) = 4√77yz

3. Destiny's work is correct so far, but she did not complete the problem. Our ending base, 12, can be simplified. Since the perfect square, 4, factors into 12, the rest of the work should look like this:

Related Links: Math algebra Dividing Square Roots Introduction to Cube Roots Simplifying Square Roots Multiplying Square Roots with Exponents Multiplying Square Roots Worksheets Dividing Square Roots Worksheets