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
The normal laws of exponents still apply, even though you now have fractions – think middle school math! Recall these exponent rules:
am an amn
am a m n , a 0 n a
abm ambm
am a bm b
m
a
amn
am
1 am
m n
New exponent rules:
n
a
m an
1 an
2 16x 5 y 4
EXAMPLE 3: Write the expression
EXAMPLE 4: Simplify the expression
1 y3
m
1 a n am
1 n
n am
a
m n
m
1 m an
1n a
5 4
using only positive exponents.
5 1 6 3 y y .
7 3 5 EXAMPLE 5: Simplify the expression x 3 y xy 6
2
EXAMPLE 6: Let k be a positive rational number. Write the expression without radicals using only positive exponents
12
a4 k
a k
1 3
Rationalizing Denominators and Numerators Before calculators, fractions with rational denominators were preferred because it was easier to estimate a value, so the transformation (called rationalizing the denominator) was taught. Now, with our technology, it’s not as necessary, BUT this skill is needed in calculus when we need to rationalize the numerator. NOTE: In the process of rationalizing the numerator or denominator, you will have a difference of two squares. EXAMPLE 7: Rationalize the denominator of each fraction: 5 a. 6
b.
3 4 5
EXAMPLE 8: Rationalize the numerator
2t h h
2t
Why do we want to be able to do this? In upcoming events, what if you wanted to find the value of this 0 fraction as h 0? Then you would get , which is indeterminate (it does NOT reduce to 1)…so you MUST 0 know this method of rationalizing the numerator!!!!