CHAPTER Radical expressions

OBJECTIVES Rewrite radical expressions by using rational exponents. Simplify and evaluate radical expressions and expressions containing rational exponents.

Rational Functions You are probably familiar with finding the square root of a number. These two operations are inverses of each other. Similarly, there are roots that correspond to larger powers. 5 and –5 are square roots of 25 because 5 2 = 25 and (–5) 2 = 25 2 is the cube root of 8 because 2 3 = 8. 2 and –2 are fourth roots of 16 because 2 4 = 16 and (–2) 4 = 16. a is the nth root of b if a n = b.

Rational Functions The nth root of a real number a can be written as the radical expression, where n is the index (plural: indices) of the radical and a is the radicand. When a number has more than one root, the radical sign indicates only the principal, or positive, root.

Example 1: Finding Real Roots Find all real roots. A. sixth roots of 64 A positive number has two real sixth roots. Because 2 6 = 64 and (–2) 6 = 64, the roots are 2 and –2. B. cube roots of –216 A positive number has two real sixth roots. Because 1 6 = 1 and (–1) 6 = 1, the roots are 1 and –1. c. cube roots of 125 A positive number has one real cube root. Because (5) 3 = 125, the root is 5.

Properties The properties of square roots also apply to nth roots.

Example 2A: Simplifying Radical Expressions Factor into perfect fourths 3  x  x  x 3x 3

Example 2B: Simplifying Radical Expressions Simplify

Example#3 Simplify the expression. Assume that all variables are positive.

Rational Exponent A rational exponent is an exponent that can be expressed as, where m and n are integers and n ≠ 0. Radical expressions can be written by using rational exponents.

Example

Rational Expressions Rational exponents have the same properties as integer exponents (See Lesson)

Example Simplify each expression

Example Simplify each expression

Chemistry Application Radium-226 is a form of radioactive element that decays over time. An initial sample of radium-226 has a mass of 500 mg. The mass of radium- 226 remaining from the initial sample after t years is given by. To the nearest milligram, how much radium-226 would be left after 800 years?

Student Guided Practice and Independent Practice Do odd problems from 2-21 in your book page 362 Then once finished do worksheet

Homework Do even numbers from in your book page 363

Closure Today we learned about rational properties Next class we are going to learn about radical functions