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. Our examples will be using the index to be 2 (square root).
Multiplying Radicals: When multiplying radicals (with the same index), multiply under the radical, and then multiply in front of the radical (any values multiplied times 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."
ANSWER:
Multiply under the radicals.
Then simplify the result.
ANSWER:
Distribute across the parentheses. Remember there is an implied "1" in front of
.
Then simplify the result.
ANSWER:
Use the distributive property to multiply. Combine like terms.
ANSWER:
Use the distributive property to multiply. There are NO like terms to be combined.
Dividing Radicals: When dividing radicals (with the same index), divide under the radical, and then divide in front of the radical (divide any values multiplied times the radicals).
ANSWER:
Divide out front and divide under the radicals.
Then simplify the result.
ANSWER:
This fraction will be in simplified form when the radical is removed from the denominator. You need to create a perfect square under the square root radical in the denominator by multiplying the top and bottom of the fraction by the same value (this is actually multiplying by "1"). The easiest approach is to multiply by the square root radical you need to convert (in this case multiply by
).
You have just "rationalized" the denominator!
Rationalize the denominator is the concept used to simplify a fraction with a square root or cube root in the denominator. It is the process of removing the root
from the denominator. You are creating a "rational" number in the denominator instead of an "irrational" number.
