# Multiply Radicals

How to multiply radicals: 2 examples and their solutions.

## Example 1

### Example

### Solution

Multiply 6x and 3x^{3}y

in a square root sign.

So √6x × √3x^{3}y = √6x⋅3x^{3}y

6 = 2⋅3

3⋅3 = 3^{2}

x⋅x^{3} = x^{1 + 3} = x^{4}

Product of Powers

Change the factors of 2⋅3^{2}⋅x^{4}⋅y to perfect squares

as much as you can.

3^{2} is already a perfect square.

x^{4} = x^{2⋅2} = (x^{2})^{2}

Power of a Power

Take the squared factors, 3 and x^{2},

out from the square root.

And leave the non-squared factors, 2 and y,

in the square root.

Simplify a Radical

So 3x^{2}√2y 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.