Copyright © Cengage Learning. All rights reserved. 3 Exponents, Polynomials and Functions

Copyright © Cengage Learning. All rights reserved. 3.1 Rules for Exponents

3 Objectives  Use the rules for exponents to simplify expressions.  Understand and use negative exponents.  Understand the relationship between rational exponents and radicals.

4 Rules for Exponents

5 To work with functions other than linear functions, we should have a good understanding of the basic rules for exponents and how to use them to simplify problems containing exponents. In this section, assume that all variables are not equal to zero. This allows us to ignore the possibility of division by zero, which is not defined.

6 Rules for Exponents The basic concept of an exponent is repeated multiplication. 2  2  2  2  2  2 = 2 6 = 64 xxxxx = x 5 xxxyy = x 3 y 2 3  3  3  3  7  7  7  7  7  7 = 3 4  7 6 = 81  =

7 Rules for Exponents Exponents allow us to write a long expression in a very compact way. When we work with exponents, there are two parts to an exponential expression: the base and the exponent. The base is the number or variable being raised to a power. The exponent is the power to which the base is being raised.

8 Rules for Exponents One of the most common operations we do with exponential expressions is to multiply them together. When we multiply any exponential expressions with the same base, the expressions can be combined into one exponential. x 7 x 2 = xxxxxxx  xx = x 9 In this example, we see that we had seven x’s multiplied by two more x’s, which gives us a total of nine x’s multiplied together.

9 Rules for Exponents Therefore, we can write a final simpler expression, x 9. Combining expressions with the same base leads us to the product rule for exponents. When more than one base is included in an expression or multiplication problem, the associative and commutative properties can be used along with the product rule for exponents to simplify the expression or multiplication.

10 Example 1 – Using the product rule for exponents Simplify the following expressions. a. x 5 x 2 x 3 b. (3f 4 g 5 )7f 2 g 7 c. (a 2 b 5 c)(a 3 b 4 c 3 ) Solution: a. x 5 x 2 x 3 = x 10 b. (3f 4 g 5 )(7f 2 g 7 ) = 3  7 f 4 f 2 g 5 g 7 = 21f 6 g 12 c. (a 2 b 5 c)(a 3 b 4 c 3 ) = a 2 a 3 b 5 b 4 cc 3 = a 5 b 9 c 4 Add the exponents. Use the commutative property to rearrange the coefficients and bases. Add the exponents of the like bases and multiply the coefficients. Add the exponents of the like bases. Use the commutative property to rearrange the bases.

11 Rules for Exponents

12 Example 2 – Using the quotient rule for exponents Simplify the following expressions. Solution: Subtract the exponents. Subtract the exponents of like bases.

13 Example 2 – Solution Subtract the exponents of like bases and reduce the coefficients. Subtract the exponents of like bases and reduce the coefficients. cont’d

14 Rules for Exponents

15 Rules for Exponents

16 Example 3 – Using the Power Rule for Exponents and Applying Exponents Simplify the following expressions. Solution: Using the power rule, multiply the exponents. Apply the exponent to each base.

17 Example 3 – Solution Using the power rule, multiply the exponents. Apply the exponent to each base and the coefficients. Apply the exponent to each base. Apply the exponent to each base and the coefficient. Using the power rule, multiply the exponents and raise the coefficient 5 to the second power. Using the power rule, multiply the exponents and raise the coefficients to the fourth power. cont’d

18 Negative Exponents and Zero as an Exponent

19 Negative Exponents and Zero as an Exponent When using the quotient rule for exponents in some situations, we will get negative numbers when we subtract the denominator’s exponent from the numerator’s exponent. A result like this makes us think about ways to define how negative exponents work. Let’s look at a division problem. Using the basic definition of exponents. Using the quotient rule for exponents.

20 Negative Exponents and Zero as an Exponent Because using the basic definition of exponents gives us and the quotient rule for exponents gives us x –2, these two expressions must be the same for both of these methods to agree and be reliable. Therefore, we have that The negative part of the exponent represents a reciprocal of the base. Notice that once we take the reciprocal of (flip) the base, the exponent becomes positive. The negative exponent only moves the base; it does not make that base negative.

21 Negative Exponents and Zero as an Exponent If a base with a negative exponent is in the denominator of a fraction, it will also be a reciprocal and will end up in the numerator of the fraction. Answers without negative exponents are easier to understand and work with, so we will write all our answers with only positive exponents.

22 Example 5 – Working with negative exponents Simplify the following expressions. Write all answers without negative exponents. Solution: The x has a negative exponent, so take the reciprocal of the base x.

23 Example 5 – Solution First, subtract the exponents. Notice that the result from subtracting exponents is always placed on top of the fraction and then moved if negative exponents remain. Move any bases with negative exponents to the bottom. cont’d The negative exponent moves the x to the bottom. The negative 2 does not move because it is not an exponent.

24 Example 5 – Solution Again, the results are in the numerator and move to the denominator if they have a negative exponent. Move any bases with negative exponents to the bottom. First, subtract the exponents. Use caution with the negatives. Subtract exponents and reduce the constants. Move any bases with negative exponents to the bottom. cont’d

25 Negative Exponents and Zero as an Exponent

26 Rational Exponents

27 All of the problems that we have worked on so far have had integer exponents. Fractions are often used as exponents in exponential problems, so we will want to know how to work with them. In this section, we will learn the basic meaning of a rational exponent and use them to simplify expressions. Rational Exponents

28 A rational exponent is another way of writing a radical such as a square root or cube root. The rational exponent represents a square root, the rational exponent represents a cube root, and so on. Rational Exponents

29 Example 7 – Rewriting rational exponents in radical form Rewrite the following exponents in radical form. Solution: a. The denominator of the exponent becomes the radical’s index. Therefore, b. The denominator of the exponent becomes the radical’s index. The numerator stays as the exponent of the variable. Therefore,

30 Example 7 – Solution c. The denominator of the exponent is 7, so the index of the radical will be 7. The 3 will stay as the exponent of the variable. Therefore, cont’d