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Radical Expressions and Rational Exponents: Lesson and Quiz

This lesson covers rewriting radical expressions with rational exponents and simplifying and evaluating them. The quiz tests understanding of the topic.

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Radical Expressions and Rational Exponents: Lesson and Quiz

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  1. 5-6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt ALgebra2

  2. 118 2. 116 75 5 3 2 5 20 Warm Up Simplify each expression. 16,807 1. 73•72 121 729 3. (32)3 4. 5.

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

  4. Essential Question • How do you multiply/divide powers with the same base when the exponents are rational?

  5. Vocabulary index rational exponent

  6. You are probably familiar with finding the square root of a number and squaring 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 52 = 25 and (–5)2 = 25 2 is the cube root of 8 because 23 = 8. 2 and –2 are fourth roots of 16 because 24 = 16 and (–2)4 = 16. a is the nth root of b if an = b.

  7. Reading Math When a radical sign shows no index, it represents a square root (index of 2). 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 unless otherwise stated.

  8. Example 1: Finding Real Roots Find all real roots. A. sixth roots of 64 A positive number has two real sixth (even) roots. Because 26 = 64 and (–2)6 = 64, the roots are 2 and –2. B. cube roots of –216 A negative number has one real cube (odd) root. Because (–6)3 = –216, the root is –6. C. fourth roots of –1024 A negative number has no real fourth (even) roots.

  9. The properties of square roots in Lesson 2-5 also apply to nth roots.

  10. Remember! When an expression contains a radical in the denominator, you must rationalize the denominator. To do so, rewrite the expression so that the denominator contains no radicals.

  11. Example 2A: Simplifying Radical Expressions Simplify each expression. Assume that all variables are positive. Factor into perfect fourths. Product Property. 3  x x x Simplify. 3x3

  12. Example 2B: Simplifying Radical Expressions Quotient Property. Simplify the numerator. ∙ Rationalize the numerator. Product Property. Simplify.

  13. 3 9 x Check It Out! Example 2c Simplify the expression. Assume that all variables are positive. Product Property of Roots. x3 Simplify.

  14. 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. The denominator of a rational exponent becomes the index of the radical. m n

  15. Write the expression (–32) in radical form and simplify. 3 5 ( ) 3 - 5 32 - 32,768 5 Example 3: 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.

  16. 5 2 4 ( ) 5 ( ) 2 4 5 4 2 2 1024 Check It Out! Example 3b 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.

  17. 4 15 1 2 8 5 A. 13 m m = = m m n n a a a a n n 3 13 Example 4: Writing Expressions by Using Rational Exponents Write each expression by using rational exponents. B. Simplify. 33 Simplify. 27

  18. 2 3 1 9 3 4 2 4 5 10 81 5 Check It Out! Example 4 Write each expression by using rational exponents. a. b. c. 103 Simplify. Simplify. Simplify 1000

  19. Rational exponents have the same properties as integer exponents.

  20. Example 5A: Simplifying Expressions with Rational Exponents Simplify each expression. Product of Powers. Simplify. 72 Evaluate the Power. 49 CheckEnter the expression in a graphing calculator.

  21. Check It Out! Example 5a Simplify each expression. Product of Powers. Simplify. 6 Evaluate the Power. Check Enter the expression in a graphing calculator.

  22. 1 4 Example 5B: Simplifying Expressions with Rational Exponents Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. Evaluate the power.

  23. Example 5B Continued Check Enter the expression in a graphing calculator.

  24. Check It Out! Example 5c Simplify each expression. Quotient of Powers. 52 Simplify. Evaluate the power. 25 Check Enter the expression in a graphing calculator.

  25. 1 1 3 3 (8) – 1 1 8 2 Check It Out! Example 5b Simplify each expression. Negative Exponent Property. Evaluate the Power.

  26. 2 3 5. Write (–216) in radical form and simplify. ( ) 2 = 36 - 3 216 5 3 5 21 21 3 Lesson Quiz: 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.

  27. Essential Question • How do you multiply/divide powers with the same base when the exponents are rational?

  28. Q: How is an artificial tree like the fourth root of −68? A: Neither has real roots.

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