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