1. Simplifying Radicals

2. Perfect Squares 64
225 1 81
256 4
100
289 9
121 16
324
144 25
400
169 36
196 49
625

3. Simplify
= 4
= 2
= 10
= 5
= 12

4. Multiply square root radicals.
For nonnegative real numbers a and b,
and
That is, the product of two square roots is the square root of
the product, and the square root of a product is the product of
the square roots.
It is important to note that the radicands not be negative numbers in the product rule. Also, in general,

5. Simplify radicals using the product rule.
A square root radical is simplified when no perfect
square factor remains under the radical sign.
This can be accomplished by using the product rule:

6. EXAMPLE 2 Simplify each radical.
Using the Product Rule to Simplify Radicals
Simplify each radical.

7. EXAMPLE 3 Find each product and simplify.
Multiplying and Simplifying Radicals
Find each product and simplify.

8. Simplify radicals by using the quotient rule.
The quotient rule for radicals is similar to the product
rule.

9. EXAMPLE 4 Simplify each radical.
Using the Quotient Rule to Simply Radicals
Simplify each radical.

10. EXAMPLE 5
Using the Quotient Rule to Divide Radicals
Simplify.

11. EXAMPLE 6
Using Both the Product and Quotient Rules
Simplify.

12. Simplify radicals involving variables.
Radicals can also involve variables.
The square root of a squared number is always nonnegative. The absolute value is used to express this.
The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers

13. EXAMPLE 7
Simplifying Radicals Involving Variables
Simplify each radical. Assume that all variables represent positive real numbers.
Solution:

14. Simplify other roots.
To simplify cube roots, look for factors that are perfect
cubes. A perfect cube is a number with a rational cube root.
For example, , and because 4 is a rational
number, 64 is a perfect cube.
For all real number for which the indicated roots exist,