Section 5-4 The Irrational Numbers Objectives: Define irrational numbers Simplify radicals Add, subtract, multiply, and divide square roots Rationalize denominators

Vocabulary A number is irrational if it can be written as a decimal that neither terminates nor repeats. The square root of a number a is the nonnegative number you have to multiply by itself to get a.

The Perfect Squares

Example 1: Simplifying Radicals Simplify each radical.

The Product Rule for Square Roots For any two positive numbers a and b,

Example 2: Multiplying Square Roots Find each product.

The Quotient Rule for Square Roots For any two positive numbers a and b,

Example 3: Using the Quotient Rule Find each quotient.

Addition and Subtraction of Like Radicals To add or subtract like radicals, add or subtract their coefficients and keep the radical the same. In symbols,

Example 4: Adding Square Roots Find the sum:

Example 5: Subtracting Square Roots Find the difference:

Example 6: Adding and Subtracting Square Roots Perform the indicated operations.

Rationalizing the Denominator It is improper to leave a square root in the denominator of a fraction. To simplify, multiply the numerator and denominator of the fraction by the square root.

Example 7: Rationalizing Denominators Simplify each radical expression.

Example 8: Simplifying the Square Root of a Fraction Simplify

Homework P. 242 #19-59 odd