Presentation on theme: "Simplifying Radicals."— Presentation transcript:

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

Simplify = 4 = 2 = 10 = 5 = 12

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,

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:

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

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

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

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

EXAMPLE 5 Using the Quotient Rule to Divide Radicals Simplify.

EXAMPLE 6 Using Both the Product and Quotient Rules Simplify.

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

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

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,