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5.5 Roots of Real Numbers and Radical Expressions

5.5 Roots of Real Numbers and Radical Expressions. Definition of n th Root. For any real numbers a and b and any positive integers n, if a n = b, then a is the n th root of b. ** For a square root the value of n is 2. radical. index. radicand. Notation.

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5.5 Roots of Real Numbers and Radical Expressions

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  1. 5.5 Roots of Real Numbers and Radical Expressions

  2. Definition of nth Root For any real numbers a and b and any positive integers n, if an = b, then a is the nth root of b. ** For a square root the value of n is 2.

  3. radical index radicand Notation Note: An index of 2 is understood but not written in a square root sign.

  4. Solution: = 3 Simplify To simplify means to find x in the equation: x4 = 81

  5. Principal square root Opposite of principal square root Both square roots Principal Root The nonnegative root of a number

  6. Summary of Roots one + root one - root no real roots even one real root, 0 one + root no - roots no + roots one - root odd

  7. Examples

  8. Examples

  9. Taking nth roots of variable expressions Using absolute value signs If the index (n) of the radical is even, the power under the radical sign is even, and the resulting power is odd, then we must use an absolute value sign.

  10. Odd Odd Even Even Even Even Examples

  11. Odd Even Even Even Even Odd 2

  12. For any numbers a and b where and , Product Property of Radicals

  13. Product Property of RadicalsExamples

  14. What to do when the index will not divide evenly into the radical???? • Smartboard Examples..\..\Algebra II Honors 2007-2008\Chapter 5\5.5 Simplifying Radicals\Simplifying Radicals.notebook

  15. Examples:

  16. Examples:

  17. For any numbers a and b where and , Quotient Property of Radicals

  18. Examples:

  19. Examples:

  20. Rationalizing the denominator Rationalizing the denominator means to remove any radicals from the denominator. Ex: Simplify

  21. Simplest Radical Form • No perfect nth power factors other than 1. • No fractions in the radicand. • No radicals in the denominator.

  22. Examples:

  23. Examples:

  24. Adding radicals We can only combine terms with radicals if we have like radicals Reverse of the Distributive Property

  25. Examples:

  26. Examples:

  27. Multiplying radicals - Distributive Property

  28. O F L I Multiplying radicals - FOIL

  29. O F L I Examples:

  30. O F L I Examples:

  31. Binomials of the form where a, b, c, d are rational numbers. Conjugates

  32. The product of conjugates is a rational number. Therefore, we can rationalize denominator of a fraction by multiplying by its conjugate.

  33. Examples:

  34. Examples:

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