SOLVING EQUATIONS INVOLVING RATIONAL
EXPONENTS
Definition:
• Rational exponent: If m and n are positive integers with m/n in lowest terms,
then
(If n is even then we require a ≥ 0.) In other words, in a
rational exponent, the numerator indicates
the power and the denominator indicates the root. For example,
Important Properties:
• To solve First,
isolate the variable. Then, raise both sides of the expression to the
reciprocal of the exponent since Finally,
solve for the variable.
• To solve Try to factor and use the zero
product property.
• Zero Product Property: If a and b are real numbers and a * b = 0, then a = 0
or b = 0.
• Whenever you raise both sides to an even power you must check your "solution"
in the original
equation. Sometimes extraneous solutions occur.
Common Mistakes to Avoid:
• Although you can raise both sides of an equation to the
same power without changing the solutions,
you can NOT raise each term to the same power.
• Remember that whenever you have the even root of a
positive number, we get two answers: one
positive and one negative. For example, if x^4 = 16 then by taking the 4th root
of both sides we get
x = 2 AND x = 2. Do NOT forget the negative answer when working with even
roots.
• Do NOT attach a when
working with odd roots. When you take the odd root of a number, you
get only one solution.
• Make sure that the variable is isolated before raising
both sides to the same power. For example,
PROBLEMS
Solve for x in each of the following equations.
First, we will isolate the variable.
Next, we will raise both sides to the 3/2
power.
Notice that we are unable to isolate the vari
able. However, we do notice that this is a
quadratictype equation. Therefore,
Setting each factor equal to zero, we obtain
OR (for an alternative way)
Letting , we get
Setting each factor equal to zero, we obtain
First, we will isolate the variable. Then we
will raise both sides to the 3/4 power.
First, we will isolate the variable.
Next, we will raise both sides to the 3/5
power.
First, we will factor this expression com
pletely.
Setting each factor equal to zero, we obtain
If we check x = 1 by substituting back into
our original equation, we find that x = 1
is a solution.
Since we cannot isolate the variable, we will
move everything to one side and factor com
pletely.
Setting each factor equal to zero, we get
Because we raised both sides to an even
power, we must check our answers in the
original equation.
Checking: x = 81 
Checking: x = 16 


Notice that the quantity containing the ra
tional exponent is already isolated. There
fore, raising both sides to the 2/3 power, we
get
Notice that although this equation does not
contain a rational exponent, to solve it we
will raise both sides to the 1/4 power.
Simplifying this last equation we get
x = 7 + 2 = 9 and x = 7  2 = 5.
Notice that we cannot solve this one by fac
toring. Therefore, we will first eliminate
the denominator of the rational exponent by
raising both sides to the 5th power.
Setting each factor equal to zero, we obtain