nth Roots
A number b is an nth root of a number a if bn = a.
The principal nth root of a real number a is
denoted 
(n ≥ 2) and defined as follows:
If n is an even number and
a ≥ 0, then 
is the nonnegative nth root;
a < 0, then 
is undefined (not a real number)
If n is an odd number, then
is the only real nth
root of a number a, and it has the same sign as a .
Note: Since is an nth
root, then 
Example: Evaluate the radical expressions.
Cancellation Rule for Exp onents and Radicals :
if n ≥ 2 and n is even
if n ≥ 3 and n is odd
Important Note:
Example: Simplify the expressions.
Rules for Radicals: (we assume that all radicals are
defined)
Example: Simplify each expression.
Simplifying Radicals:
1. Use the cancellation rule for radicals and
exponents to remove all possible factors out of the
radical.
(When it is done, the factors left in the radicand will
have smaller exponents than the index of the radical.)
2. Rationalize the denominator .
3. Reduce the index of the radical as far as possible.
Example: Simplify the expressions with radicals .
(Assume that all variables are positive when they
appear.)
Caution!
Rational Exponents
If a is real and n ≥ 2 is an integer, then
provided that 
is
defined.
Note: If n is even and a < 0, then
and 
are
not
defined.
Example: Simplify the expressions (if possible).
If a ≠ 0 is a real number and m and n are integers
containing no common factors with n ≥ 2, then
provided that 
is
defined.
Example: Evaluate each expression.
Solving Equations with Radicals/Rational
Exponents
When solving an equation linear in form with radicals
or rational exponents, simplify the equation, isolate
the most complicated radical on one side, and raise
both sides to the power equal to the index of the
radical in order to eliminate the last. You may need to
repeat this procedure if the resulting equation still
contains a radical.
Caution! If you raise both sides of an equation to
an
even power, the new equation may have more real
solutions than the original one.
Example: Equation x = 6 has solution set: {6}.
Raising both sides to the power 2, gives the equation
x^2 = 36 which has solution set x = ±6.
Thus, x = −6 is an extraneous solution to the
original equation and must be rejected.
Important! When raising to an even power, always
check each proposed solution in the original equation.
Example: Solve
Example: Solve
Note: If n is an even number,
is never
negative.
Example: Solve the equation
Arithmetic Combinations of Functions
If f and g are two functions, then
the sum f + g is a function defined by
( f + g)(x) = f (x) + g(x)
the difference f − g is the function defined by
( f − g)(x) = f (x) − g(x)
the product f * g is the function defined by
( f * g)(x) = f (x) * g(x).
The domain of each f + g , f − g , or f * g consists
of all x that are in the domains of both f and g .
The quotient f/g is the function defined by
The domain of f/g consists of all x that are in the
domains of both f and g, for which g(x) ≠ 0.
Example. Given
Find each of the fol lowing and give the domain where
it is appropriate.
