Site hosted by Angelfire.com: Build your free website today!

Text Box: Math 1314
College Algebra

LO-2
Chapter 0.3, 0.4


Skills to Master

·       Use and interpret rational exponents with 1 in the numerator

·       Use and interpret rational exponents whose numerators are not 1

·       Use and interpret radical expressions and convert between radical expressions and rational exponents

·       Simplify and combine radical expressions

·       Name a polynomial and identify its degree

·       Add, subtract, and multiply polynomial expressions


·        

Rational Exponents Whose Numerators Are 1

 

Suppose you were asked to determine the value of the numerical expression .  Not knowing the meaning of rational (or fractional) exponents, you might try applying the power-to-power rule: .  Hence you conclude that if  and  is a natural number, then  is the non-negative real number  such that .  That is,  is the nth root of .  For the above example, if , then .  That is,  is the number that, when multiplied by itself, produces 25.  Clearly, .  We see, then, that fractional exponents produce the roots of numbers.

Examples

 since .  The square root of 16 is 4

 since .  The cube root of 8 is 2

Remember that the square root of a number produces two values: one positive and one negative.  But with a rational exponent, we use the principal root, which is always positive.

 since .  The 4th root of 81 is 3 and the 4th root of   is x.  But since both  x  and –x produce the same answer, we use take the absolute value to guarantee that the result is positive.

Also, if  is even and  is negative, then there is no real number answer.  However, if  is odd and  is negative, then the answer is the negative nth root of b.

Examples

 has no real number solution since, if  is positive, there is no real number such that  (Why?)

 since

 


 

Rational Exponents Whose Numerators Are Not 1

Remember that fractions are multiplied by multiplying the numerators together and the denominators together.  That is, .  We can use this to factor fractions as well.  For example, .  Keeping this in mind and using the power-to-power rule for exponents, it is easy to see that, for example, .  Or we can write .  So for a rational expression , as an exponent it is either the nth root raised to the power of m, or as the power of m raised to the nth root.  (Note that this is the result of the commutative property of multiplication.  Why?)

In general, for real number b and whole numbers m and n, .  We can use the same line or reasoning to interpret :

 OR 

Example

Radical Expressions

Another way to express the root of a number is to use the radical sign.  The radical sign is .  In general,  and .  Here, b is called the radicand and n is the index of the expression.  For n = 2, the index is understood.  That is, .

Examples

 since

An important fact to keep in mind is that .  That is, taking the nth root of a number raised to the nth power, or raising an nth root to the nth power cancels the root and returns the radicand (or the base).  Notice that this makes sense since .  We will use this fact when solving equations that contain powers or roots.

Examples

; ;

Simplifying and Combining Radicals

For real numbers a, b and n:

This means that the multiplication of two nth roots can be written as the nth root of the product of the radicands, and the division of two nth roots can be written as the nth root of the quotient of the radicands.  Note that this only works if the values of n are the same!!!

Example

Since the roots of most numbers are irrational, we often leave a radical expression in radical form.  So rather than 1.414…, we write .  This allows us to combine the same radicals as a multiplication.  For example,  (i.e., three times the square root of seven).  In general, for real numbers a, b, c and n, .  Note that this only works when the radicals have the same value.  It is not possible, for instance, to simplify  since n and m are not equal (assuming that they are not equal).  Nor can we simplify  since a and b are not the same (assuming they are not equal).  However, if a or b can be factored such that one of the factors is a perfect power of n and the other is the same for both a and b, then we can simplify the expression.  For example,

Examples

Polynomials

A monomial is a number or the product of a number and one or more variables with whole-number exponents.  The number is called the coefficient of the monomial.

Examples

  Coefficient =

 Coefficient =

 Coefficient =

The degree of a monomial is the sum of the exponents of its variables.  All non-zero constants have a degree of zero.  For the examples above, what is the degree of each monomial?

A monomial or a sum of monomials is called a polynomial.  Each monomial in a polynomial is called a term of the polynomial.

Examples

Monomials

Binomials

Trinomials

 

The degree of a polynomial is the degree of the term with highest degree.

Any terms in a polynomial that have the same variables with the same exponents are called like terms.  The important thing to remember about like terms is that they can be added or subtracted.

Like Terms

Unlike Terms

 


 

Like terms are combined by adding or subtracting their coefficients.

Examples

Multiplying Monomials

Examples

Multiplying a Monomial and a Polynomial

Use the distributive property.

Examples

=

Multiplying Two Binomials

Use the FOIL method: First Last Inside Outside

Examples