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- 1. Lesson 2.d Simplifying Radical Expressions MC Math 02 | College and Advanced Algebra
- 2. How do you simplify a radical expression? What are the criteria for the simplest form? 1 2
- 3. Rational Exponents and Radicals 3 𝒏 𝒃 Radicals Use to denote roots Radicand Index of the radical Positive number
- 4. Rational Exponents and Radicals 4 Definition of 𝒏 𝒃 If n is a positive integer and b is a real number such that b1/n is a real number, then 𝒏 𝒃 = b1/n Index of the radical Positive number Does that mean a negative or rational index cannot exist? Consider
- 5. Rational Exponents and Radicals 5 Definition of 𝒏 𝒃 If n is a positive integer and b is a real number such that b1/n is a real number, then 𝒏 𝒃 = b1/n Any number; integer, rational, irrational, etc., 𝒏 𝝅, 𝒏 𝟏 𝟐 , etc.
- 6. Rational Exponents and Radicals 6 Definition of 𝒏 𝒃 At n=2 𝟐 𝒃 𝒐𝒓 𝒃 Principal square root of b or square root of b This is reserved for non-negative square root of b.
- 7. Rational Exponents and Radicals 7 Properties of Radicals If n and m are positive real numbers, then Product 𝒏 𝒂 ∙ 𝒏 𝒃 = 𝒏 𝒂𝒃 Quotient 𝒏 𝒂 𝒏 𝒃 = 𝒏 𝒂 𝒃 Index 𝑚 𝑛 𝑎= 𝑚𝑛 𝑎
- 8. Rational Exponents and Radicals 8 A radical is in simplest form if it meets all of the following criteria. 1. Radicand has power that is less than the index. 2. Index of radical is as small as possible. 3. Denominator is rationalized. 4. No fraction occur under the radical sign.
- 9. Rational Exponents and Radicals 9 1. The radicand contains only powers less than the index. Example: 𝒙𝟓 𝒙 𝟓 𝟐 𝒙 𝟒 𝟐 ∙ 𝒙 𝟏 𝟐 𝒙𝟐 ∙ 𝒙 𝟏 𝟐 𝒙𝟐 𝒙
- 10. Rational Exponents and Radicals 10 1. The radicand contains only powers less than the index. Example: 𝒙𝟓 𝒙𝟒 ∙ 𝒙 𝒙𝟐 𝒙
- 11. Rational Exponents and Radicals 11 1. The radicand contains only powers less than the index. Example: 𝒙𝟓 𝒙𝟐 ∙ 𝒙𝟑 𝒙 𝒙𝟑 𝒙 𝒙𝟐 ∙ 𝒙 𝒙 ∙ 𝒙 𝒙 𝒙𝟐 𝒙
- 12. Rational Exponents and Radicals 12 2. The index of radical is as small as possible. Example: 𝟗 𝒙𝟑 𝒙 𝟑 𝟗 𝒙 𝟏 𝟑 𝟑 𝒙
- 13. Rational Exponents and Radicals 13 3. The denominator has been rationalized. That is, no radicals occur in the denominator. Example: 𝟏/ 𝒙 is not in its simplest form. Rationalizing denominator will be introduced later.
- 14. Rational Exponents and Radicals 14 4. No fractions occur under the radical sign. Example: 𝟒 𝟐 𝒙𝟑
- 15. Rational Exponents and Radicals 15 Simplifying radical expressions Simplify: 𝟒 𝟑𝟐𝒙𝟑𝒚𝟒 𝟒 𝟐𝟓𝒙𝟑𝒚𝟒 Express exponentially for easy factoring. 𝟒 (𝟐𝟒𝒚𝟒) ∙ (𝟐𝒙𝟑) Factor and group factors that can be written as power of the index 𝟒 (𝟐𝟒𝒚𝟒) ∙ 𝟒 (𝟐𝒙𝟑)= 2|y| 𝟒 (𝟐𝒙𝟑) Use product property of radicals. Recall
- 16. Rational Exponents and Radicals 16 Simplifying radical expressions Simplify: 𝟑 𝟏𝟔𝟐𝒙𝟒𝒚𝟔 𝟑 (𝟐 ∙ 𝟑𝟒)𝒙𝟒𝒚𝟔 Express exponentially for easy factoring. 𝟑 𝟑𝟑𝒙𝟑𝒚𝟔 (𝟐 ∙ 𝟑𝒙) = 𝟑 𝟑𝒙𝒚𝟐 𝟑(𝟐 ∙ 𝟑𝒙) Factor and group factors that can be written as power of the index 𝟑 𝟑𝒙𝒚𝟐 𝟑 ∙ 𝟑 (𝟐 ∙ 𝟑𝒙 = 𝟑𝒙𝒚𝟐 𝟑 𝟔𝒙) Use product property of radicals. Recall
- 17. Rational Exponents and Radicals 17 Simplifying radical expressions Like radicals - have the same radicand and index 𝟑 𝟐𝒙 and −𝟒 𝟐𝒙 𝟑 𝟐𝒙 + −𝟒 𝟐𝒙 (3 + -4) 𝟐𝒙 -1 𝟐𝒙
- 18. Rational Exponents and Radicals 18 Simplifying radical expressions Combine Radical Expressions 𝟓𝒙 𝟑 𝟏𝟔𝒙𝟒 - 𝟑 𝟏𝟐𝟖𝒙𝟕 𝟓𝒙 𝟑 𝟐𝟒𝒙𝟒 - 𝟑 𝟐𝟕𝒙𝟕 Express exponentially for easy factoring. 𝟓𝒙 𝟑 𝟐𝟑𝒙𝟑 ∙ 𝟑 𝟐𝒙 - 𝟑 𝟐𝟔𝒙𝟔 ∙ 𝟑 𝟐𝒙 𝟓𝒙(𝟐𝒙) 𝟑 𝟐𝒙 - 𝟐𝟐𝒙𝟐𝟑 𝟐𝒙 Factor and group factors that can be written as power of the index Use product property of radicals. 𝟏𝟎𝒙𝟐𝟑 𝟐𝒙 - 𝟒𝒙𝟐𝟑 𝟐𝒙 6𝒙𝟐𝟑 𝟐𝒙 Simplify.
- 19. Rational Exponents and Radicals 19 Simplifying radical expressions Combine Radical Expressions (Multiplication) 𝟓 ∙ 𝟏𝟎 = 𝟓𝟎 4 𝟐𝟓 ∙ 𝟑 𝟒 = 12 𝟏𝟎𝟎
- 20. Rational Exponents and Radicals 20 Simplifying radical expressions Rationalizing Denominator 𝟑 𝟓 𝟑 𝟓 ∙ 𝟓 𝟓 = 𝟑 𝟓 ( 𝟓)𝟐