6-1 nth roots reg
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  • 1. 6-1 Nth Roots Objective: To simplify radicals having various indices, and to use a calculator to estimate the roots of numbers.
  • 2. Square Roots What power is a square root? A square is the inverse of a square root… 3 3 ?
  • 3. Square Root* Definition: For any real numbers a and b, if 2 a b then a is a square root of b or b a We can also write square roots using the ½ power. 1 2 b b
  • 4. Cube Root* Definition: For any real numbers a and b, if 3 a b then a is a cube root of b or 3 b a We can also write cube roots using the 1/3 power. 1 3 3 b b
  • 5. nth Root* Definition: For any real numbers a and b, if n a b then a is a nth root of b or n b a 1 We can also write nth roots using the power. n b 1 n n b
  • 6. Examples: Roots (of powers of 2) Even Roots: Odd Roots: 4 2 4 6 8 3 8 16 2 5 32 64 2 7 128 2 256 2 9 512 2 2 2
  • 7. Roots of negative numbers* Even roots: Negative numbers have no even roots. (undefined) Odd Roots: Negative numbers have negative roots. 4 3 27 undefined 3
  • 8. Examples: Roots (of powers of 2) Even Roots: 4 undef . Odd Roots: 3 8 2 4 16 undef . 5 32 6 64 undef . 7 128 2 9 512 2 8 256 undef . 2
  • 9. Roots: Number and Types Even Roots Positive 2 (one positive, one 1 (positive) negative) 64 Negative Odd Roots 0 (undefined) 64 3 8 undef . 64 4 1 (negative) 3 64 4
  • 10. MORE EXAMPLES if n (index) is an even integer if n is an odd integer a<0 has no real nth roots a<0 has one real nth root 2 16 a=0 has one real nth root 4i (not a realsolution) 3 8 a=0 has one real nth root 2 a>0 has two possible real nth roots a>0 has one real nth roots 40 4 x 32 4 32 30 0 24 2 0 3 27 3
  • 11. Odd Roots (of variable expressions)* When evaluating odd roots (n is odd) do not use absolute values. 3 5 a 3 3 a 15 a a 3 243 7 5 32 2
  • 12. Evaluating Roots of Monomials To evaluate nth roots of monomials: (where c is the coefficient, and x, y and z are variable expressions) n cxyz n c n 1 n x n 1 n y n 1 n z (c ) ( x ) ( y ) ( z ) 1 n or • Simplify coefficients (if possible) • For variables, evaluate each variable separately
  • 13. Evaluating Roots of Monomials* To find a root of a monomial • Split the monomial into a product of the factors, and evaluate the root of each factor. • Variables: divide the power by the root Coefficients: re-write the number as a product of prime numbers with powers, then divide the powers by the root. 49 x 8 5 32 x10 y15 49 x 8 5 25 72 5 ( x 2 )5 ( x 4 )2 5 7x4 ( y 3 )5 2x2 y3