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Graphing and Writing Linear Functions
SOLVING EQUATIONS INVOLVING RATIONAL EXPONENTS
Linear Equations and Graphing
Systems of Linear Equations
Solving Polynomial Equations
Matrix Equations and Solving Systems of Linear Equations
Introduction Part II and Solving Equations
Linear Algebra
Graphing Linear Inequalities
Using Augmented Matrices to Solve Systems of Linear Equations
Solving Linear Inequalities
Solution of the Equations
Linear Equations
Annotated Bibliography of Linear Algebra Books
Write Linear Equations in Standard Form
Graphing Linear Inequalities
Introduction to Linear Algebra for Engineers
Solving Quadratic Equations
THE HISTORY OF SOLVING QUADRATIC EQUATIONS
Systems of Linear Equations
Review for First Order Differential Equations
Systems of Nonlinear Equations & their solutions
LINEAR LEAST SQUARES FIT MAPPING METHOD FOR INFORMATION RETRIEVAL FROM NATURAL LANGUAGE TEXTS
Quadratic Equations
Syllabus for Differential Equations and Linear Alg
Linear Equations and Matrices
Solving Linear Equations
Slope-intercept form of the equation
Linear Equations
DETAILED SOLUTIONS AND CONCEPTS QUADRATIC EQUATIONS
Linear Equation Problems
Systems of Differential Equations
Linear Algebra Syllabus
Quadratic Equations and Problem Solving
LinearEquations
The Slope-Intercept Form of the Equation
Final Exam for Matrices and Linear Equations
Linear Equations

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