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**Rational Exponents and Radicals**

Section 10.5 Rational Exponents and Radicals

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**10.5 Lecture Guide: Rational Exponents and Radicals**

Objective 1: Interpret and use rational exponents.

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**The principal nth root of the real number x is denoted by either**

Examples in Radical Notation or Examples in Exponential Notation 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.

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**The principal nth root of the real number x is denoted by either**

Examples in Radical Notation or Examples in Exponential Notation Verbally For If n is even, there is no real nth root. (The nth roots will be imaginary.) is not is not a a real number. real number.

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**Use the product rule for exponents to simplify the following expressions:**

1. is the principal _________________________ root of x.

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**Use the product rule for exponents to simplify the following expressions:**

2. is the principal _________________________ root of x.

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**Use the product rule for exponents to simplify the following expressions:**

3. is the principal _________________________ root of x.

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**Represent each expression by using exponential notation, and**

evaluate each expression. 4.

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**Represent each expression by using exponential notation, and**

evaluate each expression. 5.

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**Represent each expression by using exponential notation, and**

evaluate each expression. 6.

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**Represent each expression by using exponential notation, and**

evaluate each expression. 7.

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**Represent each expression by using radical notation, and**

evaluate each expression. 8.

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**Represent each expression by using radical notation, and**

evaluate each expression. 9.

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**Represent each expression by using radical notation, and**

evaluate each expression. 10.

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**Represent each expression by using radical notation, and**

evaluate each expression. 11.

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**Represent each expression by using radical notation, and**

evaluate each expression. 12.

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**Represent each expression by using radical notation, and**

evaluate each expression. 13.

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**For a real number x and natural numbers m and n :**

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

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**For a real number x and natural numbers m and n :**

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

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**Write each expression in radical notation and evaluate.**

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**Write each expression in radical notation and evaluate.**

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**Write each expression in radical notation and evaluate.**

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**Write each expression in radical notation and evaluate.**

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**Write each expression in radical notation and evaluate.**

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**Write each expression in radical notation and evaluate.**

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**Objective 2: Use the properties of exponents.**

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You should be familiar with the properties of integer exponents from Chapter 5. Note that these properties apply to all rational exponents. Properties of Exponents Let m and n be real numbers and x, xm, xn ,y, ym, and yn be nonzero real numbers. Product rule: Product to a Power: Quotient rule: Power rule: Quotient to a Power: Negative power:

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

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

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

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

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

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

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

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

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

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

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

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

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**Simplify each expression.**

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**Simplify each expression.**

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**Simplify each expression.**

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**Simplify each expression.**

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**Use a graphing calculator or a spreadsheet to approximate**

each expression to the nearest hundredth. 36. 37. 38.

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