Rational Exponents and Radicals Definitions
Defintion 1: If x ≠ 0, then
and is
undefined
Defintion 2: If x ≠ 0, then
for any rational number n
For any positive integer n and m
Defintion 3: If n is odd, then “nth root of a” is
such that
If n is even and a > 0, then “principal nth root of a” is
such that
Defintion 4: If
is real, then
So, for odd n:
such that
for even n and a > 0:
such that
So, for odd n:
for even n:
for any a.
is

not a real number 
if n is even and a < 0 

a rational number 
if a is a perfect nth power (e.g. 9 = 3^{2} is a perfect square) 

an irrational number 
if a is not a perfect nth power 
Defintion 5: If
is real, then
Defintion 6: If
is real, then
From now on r and s are any rational numbers, m, n
and k are any intergers.
Rules
Rule 1:
Rule 2:
Rule 3:
Rule 4: For
Rule 5: For
If are real then
Rule 6:
If all indicated roots are real then
Rule 8:
Sumplifying Radicals  Rationalizing denominator
(square roots only!)
Simplifying Radicals  We will simplify
in each step as example.
Step 1: Rewrite the expression with fractional
exponents/powers.
By Rules 2, 3 and 5
Step 2: Reduce fractional powers
power = 2/3  already
reduced
power = 4/3  already
reduced
Step 3: If fractional power > 1, write it as a
mixed fraction, follow example steps.
power = 2  not
fractional power
power = 2/3 < 1
power =
Step 4: Put things back in the main problem
Step 5: Pull out factors with integer powers
Step 6: Rationalize the denomenator of the 2nd
factor
Denomenator , we need to
make it , so multiply numerator by
[We get
by solving
for z]
Step 7: Pull out the common denomenator of the
fractional power as
Step 8: Switch back to radical notation
—– Now it is fully
simplified!