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
quadratic-type 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