Multiply Radicals

How to multiply radicals: 2 examples and their solutions.

Example 1

Example

Solution

Multiply 6x and 3x3y
in a square root sign.

So √6x × √3x3y = √6x⋅3x3y

6 = 2⋅3

3⋅3 = 32
x⋅x3 = x1 + 3 = x4

Product of Powers

Change the factors of 2⋅32⋅x4⋅y to perfect squares
as much as you can.

32 is already a perfect square.
x4 = x2⋅2 = (x2)2

Power of a Power

Take the squared factors, 3 and x2,
out from the square root.

And leave the non-squared factors, 2 and y,
in the square root.

Simplify a Radical

So 3x22y is the answer.

Example 2

Example

Solution

Use the FOIL method to solve this expression.

Multiply the first two terms: √3⋅√6.
Multiply the outer terms: √3⋅(-2) = -2√3.
Multiply the inner terms: +√2⋅√6.
Multiply the last two terms: +√2⋅(-2) = -2√2.

Split √6 into √3 and √2:
6 = √3⋅√2.

3⋅√3 = (√3)2 = 3
2⋅√2 = (√2)2 = 2

Square Root

Cancel -2√3 and +2√3.
And 3√2 - 2√2 = √2.

So 3√2 - 2√3 + 2√3 - 2√2 = √2.

Add and Subtract Radicals

So √2 is the answer.