Section 9.5 Rational Exponents and Radicals

9.5 Lecture Guide: Rational Exponents and Radicals Objective: Interpret and use rational exponents.

Principle nth Root The principal nth root of the real number x is denoted by either or Verbally For The principal nth root is positive for all natural numbers n. For The principal nth root of 0 is 0. For If n is odd the principal nth root is negative. Examples in Radical Notation Examples in Exponential Notation

The principal nth root of the real number x is denoted by either or Verbally Examples in Radical Notation Examples in Exponential Notation If n is even, there is no real nth root. (The nth roots will be imaginary.) For a real number. is not real number. is not a Principle nth Root

Use the product rule for exponents to simplify the following expressions: 1. is the principal _________________________ _________________________ of x.

Use the product rule for exponents to simplify the following expressions: 2. is the principal _________________________ _________________________ of x.

Use the product rule for exponents to simplify the following expressions: 3. is the principal _________________________ _________________________ of x.

Write each expression in radical notation and evaluate. 4.

5. Write each expression in radical notation and evaluate.

6. Write each expression in radical notation and evaluate.

7.8. Write each expression in radical notation and evaluate.

9. Write each expression in radical notation and evaluate.

Rational Exponents For a real number x and rational numbers m and n : Algebraically Examples in Radical Notation Examples in Exponential Notation If is a real number* or

Rational Exponents For a real number x and rational numbers m and n : Algebraically Examples in Radical Notation Examples in Exponential Notation If is a real number* *Ifand n is even, then is not a real number.

Write each expression in radical notation and evaluate. 10.

11. Write each expression in radical notation and evaluate.

12. Write each expression in radical notation and evaluate.

13. Write each expression in radical notation and evaluate.

14. Write each expression in radical notation and evaluate.

15. Write each expression in radical notation and evaluate.

Use a graphing calculator to approximate each expression to the nearest hundredth. 16.

Use a graphing calculator to approximate each expression to the nearest hundredth. 17.

Use a graphing calculator to approximate each expression to the nearest hundredth. 18.

Use a graphing calculator to approximate each expression to the nearest hundredth. 19.

Exponent Rules You should be familiar with the properties of integer exponents. Note that these properties apply to all real exponents, including rational exponents. Let m, n, x, y, x m, x n, y m, and y n be real numbers Product rule: Product to a Power: Quotient rule: Power rule: Quotient to a Power: Negative power: with and

Simplify each expression. Assume that x is a positive real number. 20.

Simplify each expression. Assume that x is a positive real number. 21.

Simplify each expression. Assume that x is a positive real number. 22.

Simplify each expression. Assume that x is a positive real number. 23.

Simplify each expression. Assume that x and y are positive real numbers. 24.

Simplify each expression. Assume that x and y are positive real numbers. 25.

Simplify each expression. Assume that x and y are positive real numbers. 26.

Simplify each expression. Assume that x and y are positive real numbers. 27.

Simplify each expression. Assume that x and y are positive real numbers. 28.

Simplify each expression. Assume that x and y are positive real numbers. 29.

Simplify each expression. Assume that x and y are positive real numbers. 30.

Simplify each expression. Assume that x and y are positive real numbers. 31.

Simplify each expression. 32.

Simplify each expression. 33.

Simplify each expression. 34.

Simplify each expression. 35.