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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 a square root the value of n is 2. For any real numbers a and b and any positive integers n, if an = b, then a is the nth root of b.
  • 3. Definitions A perfect square is the square of a natural number. 1, 4, 9, 16, 25, and 36 are the first six perfect squares. A perfect cube is the cube of a natural number. 1, 8, 27, 64, 125, and 216 are the first six perfect cubes.
  • 4. Notation 81 4 index radical radicand Note: An index of 2 is understood but not written in a square root sign.
  • 5. Simplifying Radicals
  • 6. Simplify 81 4 To simplify means to find x in the equation: x4 = 81 Solution: = 381 4
  • 7. Principal Root The nonnegative root of a number 64 − 64 ± 64 Principal square root Opposite of principal square root Both square roots
  • 8. Summary of Roots even odd one + root one - root one + root no - roots no real roots no + roots one - root one real root, 0
  • 9. Examples ( ) 4 4 1. 169 2. - 8 -3 x x ± ( ) 22 13x= ± 2 13x= ± ( )( ) 22 8 3x= − − ( ) 2 8 3x= − −
  • 10. Examples ( ) 323 5x=3 6 3 3 3 3. 125 4. x m n− 2 5x= ( ) 33 mn= − mn= −
  • 11. 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.
  • 12. Examples ( ) ( ) 4 4 626 1. 2. an xy Even Even Even Even an= Odd 2 x y= Odd
  • 13. ( ) 6 1826 3. 4. 3 x y− Even Even 3 x= Odd Odd Even Even 2 32 3- y= ( ) 32 3- y=