Section 5.1 Rules for Exponents

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1 Objectives Section 5.1 Rules for Exponents Identify bases and exponents Multiply exponential expressions that have like bases Divide exponential expressions that have like bases Raise exponential expressions to a power Find powers of products and quotients Objective 1: Identify Bases and Exponents Recall that an exponent indicates repeated multiplication. It indicates how many times the base is used as a factor. For example, 3 5 represents the product of five 3 s. In general, we have the following definition. Natural-Number Exponents: A natural-number exponent tells how many times its base is to be used as a factor. For any number x and any natural number n, Objective 1: Identify Bases and Exponents Expressions of the form x n are called exponential expressions. The base of an exponential expression can be a number, a variable, or a combination of numbers and variables: The base is 10. The exponent is 5. Read as 10 to the fifth power or simply as 10 to the fifth. When an exponent is 1, it is usually not written. For example, 4 = 4 1 and x = x 1. 1

2 EXAMPLE 1 Identify the base and the exponent in each expression: a. 9 5, b. 7a 3, c. (7a) 3, d. t 10 Strategy To identify the base and exponent, we will look for the form x y. Why The exponent is the small raised number to the right of the base. EXAMPLE 1 a. In 9 5, the base is 9 and the exponent is 5. b. 7a 3 means 7 a 3. Thus, the base is a, not 7a. The exponent is 3. c. Because of the parentheses in (7a) 3, the base is 7a and the exponent is 3. d. Since the - symbol is not written within parentheses, the base in -t 10 is t and the exponent is 10. Objective 2: Multiply Exponential Expressions That Have Like Bases To develop a rule for multiplying exponential expressions that have the same base, we consider the product Since 6 2 means that 6 is to be used as a factor two times, and 6 3 means that 6 is to be used as a factor three times, we have: We can quickly find this result if we keep the common base of 6 and add the exponents on 6 2 and 6 3. Objective 2: Multiply Exponential Expressions That Have Like Bases This example illustrates the following rule for exponents. Product Rule for Exponents: To multiply exponential expressions that have the same base, keep the common base and add the exponents. For any number x and any natural numbers m and n, x m x n = x m + n. (Read as, x to the mth power times x to the nth power equals x to the m plus nth power. ) 2

3 EXAMPLE 4 Find an expression that represents the area of the rectangle. Geometry EXAMPLE 4 Geometry Strategy We will multiply the length of the rectangle by its width. Why The area of a rectangle is equal to the product of its length and width. The area of the rectangle is x 8 square feet, which can be written as x 8 ft 2. Objective 3: Divide Exponential Expressions That Have Like Bases To develop a rule for dividing exponential expressions that have the same base, we consider the quotient 4 5 /4 2, where the exponent in the numerator is greater than the exponent in the denominator. We can simplify this fraction by removing the common factors of 4 in the numerator and denominator: We can quickly find this result if we keep the common base, 4, and subtract the exponents on 4 5 and 4 2. Objective 3: Divide Exponential Expressions That Have Like Bases This example suggests another rule for exponents. Quotient Rule for Exponents: To divide exponential expressions that have the same base, keep the common base and subtract the exponents. For any nonzero number x and any natural numbers m and n, where m > n, x m /x n = x m n. In other words, x to the mth power divided by x to the nth power equals x to the m minus nth power. 3

4 Objective 3: Divide Exponential Expressions That Have Like Bases Recall that like terms are terms with exactly the same variables raised to exactly the same powers. To add or subtract exponential expressions, they must be like terms. To multiply or divide exponential expressions, only the bases need to be the same. EXAMPLE 5 Simplify each expression: a /20 9, b. x 9 /x 3, c. (7.5n) 12 /(7.5n) 11, d. a 3 b 8 /ab 5 Strategy In each case, we want to write an equivalent expression using each base only once. We will use the quotient rule for exponents to do this. Why The quotient rule for exponents is used to divide exponential expressions that have the same base. EXAMPLE 5 EXAMPLE 5 Read as 20 to the sixteenth power divided by 20 to the ninth power. 4

5 Objective 4: Raise Exponential Expressions to a Power To develop another rule for exponents, we consider (5 3 ) 4. Here, an exponential expression, 5 3, is raised to a power. Since 5 3 is the base and 4 is the exponent, (5 3 ) 4 can be written as Because each of the four factors of 5 3 contains three factors of 5, there are 4 3 or 12 factors of 5. We can quickly find this result if we keep the common base of 5 and multiply the exponents. Objective 4: Raise Exponential Expressions to a Power This example suggests the following rule for exponents. Power Rule for Exponents: To raise an exponential expression to a power, keep the base and multiply the exponents. For any number x and any natural numbers m and n, (x m ) n = x m n = x mn. Read as, the quantity of x to the mth power raised to the nth power equals x to the mnth power. EXAMPLE 7 EXAMPLE 7 Simplify: a. (2 3 ) 7, b. [( 6) 2 ] 5, c. (z 8 ) 8 Read as 2 cubed raised to the seventh power. Strategy In each case, we want to write an equivalent expression using one base and one exponent. We will use the power rule for exponents to do this. Read as negative six squared raised to the fifth power. Why Each expression is a power of a power. 5

6 Objective 5: Find Powers of Products and Quotients To develop more rules for exponents, we consider: the expression (2x) 3, which is a power of the product of 2 and x and the expression (2/x) 3, which is a power of the quotient of 2 and x. Objective 5: Find Powers of Products and Quotients These examples illustrate the following rules for exponents. Powers of a Product and a Quotient: To raise a product to a power, raise each factor of the product to that power. To raise a quotient to a power, raise the numerator and the denominator to that power. For any numbers x and y, and any natural number n: (xy) n = x n y n and (x/y) n = x n /y n, where y 0. EXAMPLE 9 Simplify: a. (3c) 4, b. (x 2 y 3 ) 5, c. ( ¼a 3 b) 2 EXAMPLE 9 Strategy In each case, we want to write the expression in an equivalent form in which each base is raised to a single power. We will use the power of a product rule for exponents to do this. Why Within each set of parentheses is a product, and each of those products is raised to a power. 6

7 Objective 5: Find Powers of Products and Quotients The rules for natural-number exponents are summarized as follows. If m and n represent natural numbers and there are no divisions by zero, then Section 5.2 Zero and Negative Exponents Objectives Use the zero exponent rule Use the negative integer exponent rule Use exponent rules to change negative exponents in fractions to positive exponents Use all exponent rules to simplify expressions Objective 1: Use the Zero Exponent Rule We now extend the discussion of naturalnumber exponents to include exponents that are zero and exponents that are negative integers. To develop the definition of a zero exponent, we will simplify the expression 5 3 /5 3 in two ways and compare the results. 7

8 Objective 1: Use the Zero Exponent Rule First, we apply the quotient rule for exponents, where we subtract the equal exponents in the numerator and denominator. The result is 5 0. In the second approach,we write 5 3 as and remove the common factors of 5 in the numerator and denominator. The result is 1. EXAMPLE 1 Simplify. Assume a 0: a. ( 8) 0, b. (14/15) 0, c. (3a) 0, d. 3a 0 Strategy We note that each exponent is 0. To simplify the expressions, we will identify the base and use the zeroexponent rule. Since 5 3 /5 3 = 5 0 and 5 3 /5 3 = 1, we conclude that 5 0 = 1. This observation suggests the following definition: Zero Exponents: Any nonzero base raised to the 0 power is 1. Why If an expression contains a nonzero base raised to the 0 power, we can replace it with 1. For any nonzero real number x, x 0 = 1. Read as x to the zero power equals 1. EXAMPLE 1 Objective 2: Use the Negative Integer Exponent Rule To develop the definition of a negative exponent, we will simplify 6 2 /6 5 in two ways and compare the results. If we apply the quotient rule for exponents, where we subtract the greater exponent in the denominator from the lesser exponent in the numerator, we get 6 3. In the second approach, we remove the two common factors of 6 to get 1/6 3. Since 6 2 /6 5 = 6 3 and 6 2 /6 5 = 1/6 3, we conclude that 6 3 = 1/6 3. Note that 6 3 is equal to the reciprocal of 6 3. This observation suggests the following definition: Negative Exponents: For any nonzero real number x and any integer n, x n = 1/x n. In words, x n is the reciprocal of x n. 8

9 EXAMPLE 2 Express using positive exponents and simplify, if possible: a. 3 2, b. y 1, c. ( 2) 3, d Strategy Since each exponent is a negative number, we will use the negative exponent rule. EXAMPLE 2 Why This rule enables us to write an exponential expression that has a negative exponent in an equivalent form using a positive exponent. Objective 3: Use Exponent Rules to Change Negative Exponents in Fractions to Positive Exponents Negative exponents can appear in the numerator and/or the denominator of a fraction. To develop rules for such situations, we consider the following example. We can obtain this result in a simpler way. In a 4 /b 3, we can move a 4 from the numerator to the denominator and change the sign of the exponent, and we can move b 3 from the denominator to the numerator and change the sign of the exponent. Objective 3: Use Exponent Rules to Change Negative Exponents in Fractions to Positive Exponents This example suggests the following rules. Changing from Negative to Positive Exponents: A factor can be moved from the denominator to the numerator or from the numerator to the denominator of a fraction if the sign of its exponent is changed. For any nonzero real numbers x and y, and any integers m and n, 1/x n = x n, and x m /y n = y n /x m. These rules streamline the process when simplifying fractions involving negative exponents. 9

10 Objective 3: Use Exponent Rules to Change Negative Exponents in Fractions to Positive Exponents When a fraction is raised to a negative power, we can use rules for exponents to change the sign of the exponent. For example, we see that: This process can be streamlined using the following rule. Negative Exponents and Reciprocals: A fraction raised to a power is equal to the reciprocal of the fraction raised to the opposite power. For any nonzero real numbers x and y, and any integer n, (x/y) n = (y/x) n. EXAMPLE 5 Simplify: (4/m) 2 Strategy We want to write this fraction that is raised to a negative power in an equivalent form that involves a positive power. We will use the negative exponent and reciprocal rules to do this. Why It is usually easier to simplify exponential expressions if the exponents are positive. EXAMPLE 5 Objective 4: Use All Exponent Rules to Simplify Expressions The rules for exponents involving products, powers, and quotients are also true for zero and negative exponents. 10

11 Objective 4: Use All Exponent Rules to Simplify Expressions The rules for exponents are used to simplify expressions involving products, quotients, and powers. In general, an expression involving exponents is simplified when: Each base occurs only once. No powers are raised to powers There are no parentheses. There are no negative or zero exponents. EXAMPLE 6 Simplify. Do not use negative exponents in the answer. a. x 5 x 3, b. x 3 /x 7, c. (x 3 ) 2, d. (2a 3 b 5 ) 3, e. (3/b 5 ) 4 Strategy In each case, we want to write an equivalent expression using one base and one positive exponent. We will use rules for exponents to do this. Why These expressions are not in simplest form. In parts a and b, the base x occurs more than once. In parts c, d, and e, there is a negative exponent. EXAMPLE 6 EXAMPLE 6 11

12 Objectives Section 5.3 Scientific Notation Convert from scientific to standard notation Write numbers in scientific notation Perform computations with scientific notation Objective 1: Convert from Scientific to Standard Notation Scientists often deal with extremely large and small numbers. For example, the distance from the Earth to the sun is approximately 150,000,000 kilometers. The influenza virus, which causes flu symptoms of cough, sore throat, and headache, has a diameter of inch. Objective 1: Convert from Scientific to Standard Notation The numbers 150,000,000 and are written in standard notation, which is also called decimal notation. Because they contain many zeros, they are difficult to read and cumbersome to work with in calculations. In this section, we will discuss a more convenient form in which we can write such numbers. 12

13 Objective 1: Convert from Scientific to Standard Notation Scientific notation provides a compact way of writing very large or very small numbers. A positive number is written in scientific notation when it is written in the form N 10 n, where 1 N < 10 and n is an integer. To write numbers in scientific notation, you need to be familiar with powers of 10, like those listed in the table below. Objective 1: Convert from Scientific to Standard Notation Two examples of numbers written in scientific notation are shown below. Note that each of them is the product of a decimal number (between 1 and 10) and a power of 10. A number written in scientific notation can be converted to standard notation by performing the indicated multiplication. Objective 1: Convert from Scientific to Standard Notation For example, to convert , we recall that multiplying a decimal by 100 moves the decimal point 2 places to the right. Objective 1: Convert from Scientific to Standard Notation In and , the exponent gives the number of decimal places that the decimal point moves, and the sign of the exponent indicates the direction in which it moves. Applying this observation to several other examples, we have: To convert to standard notation, we recall that dividing a decimal by 1,000 moves the decimal point 3 places to the left. 13

14 Objective 1: Convert from Scientific to Standard Notation The following procedure summarizes our observations. Converting from Scientific to Standard Notation: 1. If the exponent is positive, move the decimal point the same number of places to the right as the exponent. 2. If the exponent is negative, move the decimal point the same number of places to the left as the absolute value of the exponent. EXAMPLE 1 Convert to standard notation: a , b Strategy In each case, we need to identify the exponent on the power of 10 and consider its sign. Why The exponent gives the number of decimal places that we should move the decimal point. The sign of the exponent indicates whether it should be moved to the right or the left. EXAMPLE 1 a. Since the exponent in 10 5 is 5, the decimal point moves 5 places to the right. Thus, = 346,700. b. Since the exponent in 10 4 is 4, the decimal point moves 4 places to the left. Objective 2: Write Numbers in Scientific Notation To write a number in scientific notation (N 10 n ) we first determine N and then n. Note: The results from the next example illustrate the following forms to use when converting numbers from standard to scientific notation. Thus, =

15 EXAMPLE 2 Write each number in scientific notation: a. 150,000,000, b , c Strategy We will write each number as the product of a number between 1 and 10 and a power of 10. EXAMPLE 2 a. We must write 150,000,000 (the distance in kilometers from the Earth to the sun) as the product of a number between 1 and 10 and a power of 10. We note that 1.5 lies between 1 and 10. To obtain 150,000,000, we must move the decimal point in 1.5 exactly 8 places to the right. This will happen if we multiply 1.5 by Therefore, Why Numbers written in scientific notation have the form N 10 n. EXAMPLE 2 b. We must write (the diameter in inches of a flu virus) as the product of a number between 1 and 10 and a power of 10. We note that 2.56 lies between 1 and 10. To obtain , the decimal point in 2.56 must be moved 6 places to the left. EXAMPLE 2 c. The number is not written in scientific notation because 432 is not a number between 1 and 10. To write this number in scientific notation, we proceed as follows: This will happen if we multiply 2.56 by Therefore, Written in scientific notation, is

16 Objective 3: Perform Calculations with Scientific Notation Another advantage of scientific notation becomes apparent when we evaluate products or quotients that involve very large or small numbers. If we express those numbers in scientific notation, we can use rules for exponents to make the calculations easier. EXAMPLE 3 Astronomy Except for the sun, the nearest star visible to the naked eye from most parts of the United States is Sirius. Light from Sirius reaches Earth in about 70,000 hours. If light travels at approximately 670,000,000 mph, how far from Earth is Sirius? Strategy We can use the formula d = rt to find the distance from Sirius to Earth. Why We know the rate at which light travels and the time it takes to travel from Sirius to the Earth. EXAMPLE 3 Astronomy The rate at which light travels is 670,000,000 mph and the time it takes the light to travel from Sirius to Earth is 70,000 hr. To find the distance from Sirius to Earth, we proceed as follows: EXAMPLE 3 Astronomy We note that 46.9 is not between 0 and 10, so is not written in scientific notation. To answer in scientific notation, we proceed as follows. Conclusion: Sirius is approximately or 46,900,000,000,000 miles from Earth. 16