Multiplying and dividing radicals makes use of the "Product Rule" and the "Quotient Rule" as seen at the right. The "n" simply means that the index could be any value. The examples on this page use square and cube roots.
 Multiplying Radicals When multiplying radicals (with the same index), multiply under the radicals, and then multiply any values directly in front of the radicals.

 ANSWER: Multiply the values under the radicals. Then simplify the result. ANSWER: Multiply out front and multiply under the radicals. Then simplify the result.
 Product Rule where a ≥ 0, b≥ 0 "The radical of a product is equal to the product of the radicals of each factor."

 Quotient Rule where a ≥ 0, b > 0 "The radical of a quotient is equal to the quotient of the radicals of the numerator and denominator."

Then simplify the result.

Distribute across the parentheses. Remember there is an implied "1" in front of .

Then simplify the result.

Use the distributive property to multiply. Combine like terms.

Use the distributive property to multiply. There are NO like terms to be combined.

In this problem, it is easier to reduce the radicals before multiplying since the perfect cube (27) can be more clearly seen in each radicand. Yes, you could have chosen, instead, to multiply and then reduce.

 A conjugate is a binomial formed by negating the second term of a binomial. Example: (x + y) and (x - y) are conjugates. A conjugate involving an imaginary number is called a complex conjugate. Example: (a + bi) and (a - bi) are complex conjugates

These terms are conjugates involving a radical. As with all conjugates, when multiplied, the middle terms cancel each other out. Notice the squaring of the square root. Notice that when the conjugates were multiplied, the radicals disappeared!

 Dividing Radicals When dividing radicals (with the same index), divide under the radical, and then divide the values directly in front of the radical.