Exponents and Radicals

    Usually, there are many ways to solve a problem in algebra. The more fluency you have with exponentials and radicals the more options you have in your procedure of simplifying algebraic expressions.

    Repeated multiplication can be written in exponential form. For example, if is a real number then is multiplied out times. The number is called the base is called the exponent. One of the most common beginner mistakes is to not distinguish between and   

Definition (Zero and Negative Exponents) If is a non-zero real number, then
    
    (i)

    (ii)

Proposition (Properties of Exponents) If and are real numbers and if and are integers, then



provided these numbers are defined.

Proposition (Negative Exponents) If and are real numbers and, and are integers, then
    
    (i)

    (ii)
    
provided these numbers are defined.

Example (Properties of Exponents) Simplify the following expressions.

(a)

    Solution. We find that,

.


(b)

    Solution. We find that,

.


(c)

    Solution. We find that,

.

Properties of Radicals

    Integers such as are called perfect squares. Similarly, integers such as are called perfect cubes.

Definition (Principal nth Root) Let be a positive integer greater than 1, and let be a real number.
    
    (i) If then
    
    (ii) If then is the positive real number such that
    
    (iii) If and is odd, then is the negative real number such that
    
    (iv) If and is even, then is not a real number.

    The expression is called a radical, the number is called the radicand, and is the index of the radical. The symbol is called the radical sign.

Proposition (Principal nth Root) Let be a positive integer greater than 1, and let be a real number.
    
    (i) If is a real number, then .
    
    (ii) If then
    
    (iii) If and is odd, then
    
    (iv) If and is even, then
    

Proposition (Properties of Radicals) Let be a positive integer greater than 1, and let be a real number. Then,
    
    (i)
    
    (ii)
    
    (iii)
    
provided the indicated roots exist.

    To simplify the radical means to remove factors from the radical until no factor in the radicand has an exponent greater than or equal to in the index of the radical and the index is as low as possible.

Example (Properties of Radicals) Assuming the following are real numbers; simplify.

(a)

    Solution. We find that,




(b)

    Solution. We find that,
    



(c)

    Solution. We find that,

Definition (Rationalizing the Dominator) If the denominator contains a factor of the form with and then multiplying numerator and denominator by will eliminate the radical from the denominator, since The process of eliminating radicals from the denominator is called rationalizing the denominator.

Definition (Simplify a Radical)    An expression involving radicals is in simplest form when the following conditions are satisfied:

    (i) All possible factors have been removed form the radical.
    
    (ii) All fractions have radical-free denominators (accomplished by rationalizing the denominator).
    
    (iii) The index of the radical is reduced.

Definition (Conjugates of Radicals) To rationalize a denominator or numerator of the form or multiply both numerator and denominator by a conjugate: and are conjugates of each other. For cube roots, choose a rationalizing factor that generates a perfect cube.

Example (Rationalizing the Denominator) Rationalize the denominator and simplify.

(a)

    Solution. We use to rationalize the denominator, as follows,



(b)

    Solution. We use to rationalize the denominator, as follows,
    



(c)

    Solution. We use to rationalize the denominator, as follows,
    

Definition (Rationalizing the Numerator) If the numerator contains a factor of the form with and then multiplying numerator and denominator by will eliminate the radical from the numerator, since The process of eliminating radicals from the numerator is called rationalizing the numerator.

Example (Rationalizing the Numerator) Rationalize the numerator and simplify.

(a)

    Solution. We use to rationalize the numerator, as follows,
    



(b)

    Solution. We use to rationalize the numerator, as follows,




(c)

    Solution. We use to rationalize the numerator, as follows,

Rational Exponents

    Rational exponent can be difficult for the beginner. For example, is not the same as since but is not a real number because is not defined.

Definition (Rational Exponents) Let be a rational number, where is a positive integer greater than If is a real number such that exists, then

    (i)
    
    (ii)
    
    (iii)

Example (Rational Exponents) Simplify the following expressions.

(a)

    Solution. We find that,










(b)

    Solution. We find that,
    




(c)

    Solution. We find that,
    

• print this page
• pages linking here
• related pages