/5.1__Radicals_and_Rational_Exponents

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5.1 – Radicals and Rational Exponents ☼ Recall nth roots. These are the solutions to x n  c . Depending on whether n is odd or even, and whether c is positive or negative, the equation may have two, one, or no solutions as shown below: n is odd n is even exactly one solution c  0 , one positive c  0 , one solution, c  0 , no solution for any c & one negative solution x 0 y y y y y=c

y=c x

x

y=c

x

x y=c

nth Roots: ☼ If c is a real number and n a positive integer, the nth root of c is denoted by one of the following: n

1 n

c or c , and is defined to be:

 the solution of x n  c when n is odd, OR  the nonnegative solution of x n  c when n is even and c  0

EXAMPLE 1: Simplify each expression: 8 a. 6

c.

3

45  125

b.



d. 6  a 6  a , a  0

1000x 3y 7

EXAMPLE 2: Use a calculator to approximate each expression to the nearest ten-thousandths: a.

1 35 8

b.

1 12 285

Definition of Rational Exponents t Let c be a positive real number and let be a rational number with positive denominator: k t k c

is defined to be the number

1 t k c

 

t

 1  ck      

k

ct 

 c k

t


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