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6-1. Radical Expressions and Rational Exponents. Warm Up. Lesson Presentation. Lesson Quiz. Holt ALgebra2. 6-1. Objectives. Rewrite radical expressions by using rational exponents. Simplify and evaluate radical expressions and expressions containing rational exponents. 6-1. Vocabulary.

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6-1

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  1. 6-1 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt ALgebra2

  2. 6-1 Objectives Rewrite radical expressions by using rational exponents. Simplify and evaluate radical expressions and expressions containing rational exponents.

  3. 6-1 Vocabulary index rational exponent

  4. Writing Math The denominator of a rational exponent becomes the index of the radical. 6-1

  5. Write the expression (–32) in radical form and simplify. 3 5 ( ) 3 - 5 32 - 32,768 5 6-1 Example 1: Writing Expressions in Radical Form Method 1 Evaluate the root first. Method 2 Evaluate the power first. Write with a radical. Write with a radical. (–2)3 Evaluate the root. Evaluate the power. –8 Evaluate the power. –8 Evaluate the root.

  6. 1 3 64 ( ) 1 ( ) 3 64 1 64 3 3 64 6-1 Check It Out! Example 1a Write the expression in radical form, and simplify. Method 1 Evaluate the root first. Method 2 Evaluate the power first. Write with a radical. Write will a radical. (4)1 Evaluate the root. Evaluate the power. 4 Evaluate the power. 4 Evaluate the root.

  7. 5 2 4 ( ) 5 ( ) 2 4 5 4 2 2 1024 6-1 Check It Out! Example 1b Write the expression in radical form, and simplify. Method 1 Evaluatethe root first. Method 2 Evaluatethe power first. Write with a radical. Write with a radical. (2)5 Evaluate the root. Evaluate the power. 32 Evaluate the power. 32 Evaluate the root.

  8. 3 4 625 ( ) 3 ( ) 3 4 625 625 4 244,140,625 4 6-1 Check It Out! Example 1c Write the expression in radical form, and simplify. Method 1 Evaluatethe root first. Method 2 Evaluate the power first. Write with a radical. Write with a radical. (5)3 Evaluate the root. Evaluate the power. 125 Evaluate the power. 125 Evaluate the root.

  9. 4 15 1 8 5 2 A. B. 13 m m = = m m n n a a a a n n 3 13 6-1 Example 2: Writing Expressions by Using Rational Exponents Write each expression by using rational exponents. Simplify. 33 Simplify. 27

  10. 2 3 1 9 3 4 2 4 5 10 81 5 6-1 Check It Out! Example 3 Write each expression by using rational exponents. a. b. c. 103 Simplify. Simplify. 1000

  11. 6-1 Rational exponents have the same properties as integer exponents (See Lesson 1-5)

  12. 6-1 Example 4: Simplifying Expressions with Rational Exponents Simplify each expression. Product of Powers. Simplify. 72 Evaluate the Power. 49 Older TI Program New TI Program CheckEnter the expression in a graphing calculator.

  13. 1 4 6-1 Example 5: Simplifying Expressions with Rational Exponents Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. Evaluate the power.

  14. 6-1 Example 5 Continued Check Enter the expression in a graphing calculator. New TI Program Older TI Program

  15. 6-1 Check It Out! Example 6 Simplify each expression. Product of Powers. Simplify. 6 Evaluate the Power. Check Enter the expression in a graphing calculator.

  16. 1 1 3 3 (–8)– 1 – 1 –8 2 6-1 Check It Out! Example 7 Simplify each expression. Negative Exponent Property. Evaluate the Power. Check Enter the expression in a graphing calculator.

  17. 6-1 Check It Out! Example 8 Simplify each expression. Quotient of Powers. 52 Simplify. Evaluate the power. 25 Check Enter the expression in a graphing calculator.

  18. 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? 6-1 Example 9: Chemistry Application

  19. 500 (200–) = 500(2– ) 1 1 800 1 t 2 2 1600 1600 2 = 500(2– ) = 500() 500 1 2 2 = 6-1 Example 9 Continued Substitute 800 for t. Simplify. Negative Exponent Property. Simplify. Use a calculator. ≈ 354 The amount of radium-226 left after 800 years would be about 354 mg.

  20. To find the distance a fret should be place from the bridge on a guitar, multiply the length of the string by , where n is the number of notes higher that the string’s root note. Where should the fret be placed to produce the E note that is one octave higher on the E string (12 notes higher)? 6-1 Check It Out! Music Application Use 64 cm for the length of the string, and substitute 12 for n. = 64(2–1) Simplify. Negative Exponent Property. Simplify. = 32 The fret should be placed 32 cm from the bridge.

  21. 2 3 5. Write (–216) in radical form and simplify. ( ) 2 = 36 - 3 216 5 3 5 21 21 3 6-1 Lesson Exit Ticket: Part I Find all real roots. 1. fourth roots of 625 –5, 5 2. fifth roots of –243 –3 Simplify each expression. 4. 4y2 8 3. 256 y 4 2 6. Write using rational exponents.

  22. 6-1 Lesson Exit Ticket: Part II 7. If $2000 is invested at 4% interest compounded monthly, the value of the investment after t years is given by . What is the value of the investment after 3.5 years? $2300.01

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