m81342 Integer Exponents and Scientific Notation By the end of this section, you will be able to: Use the definition of a negative exponent Simplify expressions with integer exponents Convert from decimal notation to scientific notation Convert scientific notation to decimal form Multiply and divide using scientific notation d741bac6-bd7f-4e82-9df0-bd1e094380b9 Before you get started, take this readiness quiz. What is the place value of the 6 in the number 64,891? If you missed this problem, review . ten thousand Name the decimal 0.0012. If you missed this problem, review . twelve ten-thousandths Subtract: 5−(−3). If you missed this problem, review . 8

The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger. If a is a real number, a≠0, and m,n are whole numbers, then aman=am−n,m>nandaman=1an−m,n>m What if we just subtract exponents, regardless of which is larger? Let’s consider x2x5. We subtract the exponent in the denominator from the exponent in the numerator. x2x5 x2−5 x−3 We can also simplify x2x5 by dividing out common factors: x2x5. This implies that x−3=1x3 and it leads us to the definition of a negative exponent. If n is a positive integer and a≠0, then a−n=1an. The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents. Simplify: ⓐ 4−2ⓑ 10−3 ⓐ 4−2 Use the definition of a negative exponent, a−n=1an. 142 Simplify. 116 ⓑ 10−3 Use the definition of a negative exponent, a−n=1an. 1103 Simplify. 11000 Simplify: ⓐ 2−3ⓑ 10−2 ⓐ 18ⓑ 1100 Simplify: ⓐ 3−2ⓑ 10−4 ⓐ 19ⓑ 110,000 When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent. Simplify: ⓐ (−3)−2ⓑ −3−2 The negative in the exponent does not affect the sign of the base. ⓐ The exponent applies to the base, −3. (−3)−2 Take the reciprocal of the base and change the sign of the exponent. 1(−3)2 Simplify. 19 ⓑ The expression −3−2 means "find the opposite of 3−2". The exponent applies only to the base, 3. −3−2 Rewrite as a product with −1. −1·3−2 Take the reciprocal of the base and change the sign of the exponent. −1·132 Simplify. −19 Simplify: ⓐ (−5)−2ⓑ −5−2 ⓐ 125ⓑ −125 Simplify: ⓐ (−2)−2ⓑ −2−2 ⓐ 14ⓑ −14 We must be careful to follow the order of operations. In the next example, parts ⓐ and ⓑ look similar, but we get different results. Simplify: ⓐ 4·2−1ⓑ (4·2)−1 Remember to always follow the order of operations. ⓐ Do exponents before multiplication. 4·2−1 Use a−n=1an. 4·121 Simplify. 2 ⓑ (4·2)−1 Simplify inside the parentheses first. (8)−1 Use a−n=1an. 181 Simplify. 18 Simplify: ⓐ 6·3−1ⓑ (6·3)−1 ⓐ 2ⓑ 118 Simplify: ⓐ 8·2−2ⓑ (8·2)−2 ⓐ 2ⓑ 1256 When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. Simplify: x−6. x−6 Use the definition of a negative exponent, a−n=1an. 1x6 Simplify: y−7. 1y7 Simplify: z−8. 1z8 When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example. Simplify: ⓐ 5y−1ⓑ (5y)−1ⓒ (−5y)−1 ⓐ Notice the exponent applies to just the base y. 5y−1 Take the reciprocal of y and change the sign of the exponent. 5·1y1 Simplify. 5y ⓑ Here the parentheses make the exponent apply to the base 5y. (5y)−1 Take the reciprocal of 5y and change the sign of the exponent. 1(5y)1 Simplify. 15y ⓒ (−5y)−1 The base is −5y. Take the reciprocal of −5y and change the sign of the exponent. 1(−5y)1 Simplify. 1−5y Use a−b=−ab. −15y Simplify: ⓐ 8p−1ⓑ (8p)−1ⓒ (−8p)−1 ⓐ 8pⓑ 18pⓒ −18p Simplify: ⓐ 11q−1ⓑ (11q)−1ⓒ (−11q)−1 ⓐ 11qⓑ 111qⓒ −111q Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, aman=am−n, where a≠0 and m and n are integers. When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, a−n=1an.

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference. If a,b are real numbers and m,n are integers, then Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power Property(ab)m=ambmQuotient Propertyaman=am−n,a≠0Zero Exponent Propertya0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0Definition of Negative Exponenta−n=1an Simplify: ⓐ x−4·x6ⓑ y−6·y4ⓒ z−5·z−3 ⓐ x−4·x6 Use the Product Property, am·an=am+n. x−4+6 Simplify. x2 ⓑ y−6·y4 The bases are the same, so add the exponents. y−6+4 Simplify. y−2 Use the definition of a negative exponent, a−n=1an. 1y2 ⓒ z−5·z−3 The bases are the same, so add the exponents. z−5−3 Simplify. z−8 Use the definition of a negative exponent, a−n=1an. 1z8 Simplify: ⓐ x−3·x7ⓑ y−7·y2ⓒ z−4·z−5 ⓐ x4ⓑ 1y5ⓒ 1z9 Simplify: ⓐ a−1·a6ⓑ b−8·b4ⓒ c−8·c−7 ⓐ a5ⓑ 1b4ⓒ 1c15 In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents. Simplify: (m4n−3)(m−5n−2). (m4n−3)(m−5n−2) Use the Commutative Property to get like bases together. m4m−5·n−2n−3 Add the exponents for each base. m−1·n−5 Take reciprocals and change the signs of the exponents. 1m1·1n5 Simplify. 1mn5 Simplify: (p6q−2)(p−9q−1). 1p3q3 Simplify: (r5s−3)(r−7s−5). 1r2s8 If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Use Multiplication Properties of Exponents. Simplify: (2x−6y8)(−5x5y−3). (2x−6y8)(−5x5y−3) Rewrite with the like bases together. 2(−5)·(x−6x5)·(y8y−3) Simplify. −10·x−1·y5 Use the definition of a negative exponent, a−n=1an. −10·1x1·y5 Simplify. −10y5x Simplify: (3u−5v7)(−4u4v−2). −12v5u Simplify: (−6c−6d4)(−5c−2d−1). 30d3c8 In the next two examples, we’ll use the Power Property and the Product to a Power Property. Simplify: (k3)−2. (k3)−2 Use the Product to a Power Property, (ab)m=ambm. k3(−2) Simplify. k−6 Rewrite with a positive exponent. 1k6 Simplify: (x4)−1. 1x4 Simplify: (y2)−2. 1y4 Simplify: (5x−3)2. (5x−3)2 Use the Product to a Power Property, (ab)m=ambm. 52(x−3)2 Simplify 52 and multiply the exponents of x using the Power Property, (am)n=am·n. 25x−6 Rewrite x−6 by using the definition of a negative exponent, a−n=1an. 25·1x6 Simplify 25x6 Simplify: (8a−4)2. 64a8 Simplify: (2c−4)3. 8c12 To simplify a fraction, we use the Quotient Property. Simplify: r5r−4. Use the Quotient Property, aman=am−n. Simplify. Simplify: x8x−3. x¹¹ Simplify: y7y−6. y¹³

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4000 and 0.004. We know that 4000 means 4×1000 and 0.004 means 4×11000. If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way: 40000.0044×10004×110004×1034×11034×10−3When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation. A number is expressed in scientific notation when it is of the form a×10n where a≥1 and a<10 and n is an integer. It is customary in scientific notation to use × as the multiplication sign, even though we avoid using this sign elsewhere in algebra. Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier. If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation. In both cases, the decimal was moved 3 places to get the first factor, 4, by itself. The power of 10 is positive when the number is larger than 1:4000=4×103. The power of 10 is negative when the number is between 0 and 1:0.004=4×10−3. Write 37,000 in scientific notation. Step 1: Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Step 2: Count the number of decimal places, n, that the decimal point was moved. 3.700004 places Step 3: Write the number as a product with a power of 10. 3.7×104 If the original number is: greater than 1, the power of 10 will be 10n. between 0 and 1, the power of 10 will be 10−n Step 4: Check. 104 is 10,000 and 10,000 times 3.7 will be 37,000. 37,000=3.7×104 Write in scientific notation: 96,000. 9.6 × 10⁴ Write in scientific notation: 48,300. 4.83 × 10⁴ Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Count the number of decimal places, n, that the decimal point was moved. Write the number as a product with a power of 10. If the original number is: greater than 1, the power of 10 will be 10n. between 0 and 1, the power of 10 will be 10−n. Check. Write in scientific notation: 0.0052. 0.0052 Move the decimal point to get 5.2, a number between 1 and 10. Count the number of decimal places the point was moved. 3 places Write as a product with a power of 10. 5.2 ×10−3 Check your answer:5.2×10−35.2×11035.2×110005.2×0.0010.0052 0.0052=5.2×10−3 Write in scientific notation: 0.0078. 7.8 × 10⁻³ Write in scientific notation: 0.0129. 1.29 × 10⁻²

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see. 9.12×1049.12×10−49.12×10,0009.12×0.000191,2000.000912If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form. In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left. Convert to decimal form: 6.2×103. Step 1: Determine the exponent, n, on the factor 10. 6.2×103 Step 2: Move the decimal point n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point |n| places to the left. 6,200 Step 3: Check to see if your answer makes sense. 103 is 1000 and 1000 times 6.2 will be 6,200. 6.2 ×103=6,200 Convert to decimal form: 1.3×103. 1,300 Convert to decimal form: 9.25×104. 92,500 Determine the exponent, n, on the factor 10. Move the decimal n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point |n| places to the left. Check. Convert to decimal form: 8.9×10−2. 8.9 ×10−2 Determine the exponent n, on the factor 10. The exponent is −2. Move the decimal point 2 places to the left. Add zeros as needed for placeholders. 0.089 8.9×10−2=0.089 The Check is left to you. Convert to decimal form: 1.2×10−4. 0.00012 Convert to decimal form: 7.5×10−2. 0.075

We use the Properties of Exponents to multiply and divide numbers in scientific notation. Multiply. Write answers in decimal form: (4×105)(2×10−7). (4×105)(2×10−7) Use the Commutative Property to rearrange the factors. 4·2·105·10−7 Multiply 4 by 2 and use the Product Property to multiply 105 by 10−7. 8×10−2 Change to decimal form by moving the decimal two places left. 0.08 Multiply. Write answers in decimal form: (3×106)(2×10−8). 0.06 Multiply. Write answers in decimal form: (3×10−2)(3×10−1). 0.009 Divide. Write answers in decimal form: 9×1033×10−2. 9×1033×10−2 Separate the factors. 93×10310−2 Divide 9 by 3 and use the Quotient Property to divide 103 by 10−2. 3×105 Change to decimal form by moving the decimal five places right. 300,000 Divide. Write answers in decimal form: 8×1042×10−1. 400,000 Divide. Write answers in decimal form: 8×1024×10−2. 20,000 Negative Exponents Examples of Simplifying Expressions with Negative Exponents Scientific Notation

Summary of Exponent Properties If a,b are real numbers and m,n are integers, then Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power Property(ab)m=ambmQuotient Propertyaman=am−n,a≠0Zero Exponent Propertya0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0Definition of Negative Exponenta−n=1an Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation: Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Count the number of decimal places, n, that the decimal point was moved. Write the number as a product with a power of 10. If the original number is greater than 1, the power of 10 will be 10n. If the original number is between 0 and 1, the power of 10 will be 10-n. Check. Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form: Determine the exponent, n, on the factor 10. Move the decimal n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point |n| places to the left. Check.

Use the Definition of a Negative Exponent In the following exercises, simplify. 5−3 8−2 164 3−4 2−5 132 7−1 10−1 110 2−3+2−2 3−2+3−1 49 3−1+4−1 10−1+2−1 35 100−10−1+10−2 20−2−1+2−2 34 ⓐ (−6)−2ⓑ −6−2 ⓐ (−8)−2ⓑ −8−2 ⓐ 164ⓑ −164 ⓐ (−10)−4ⓑ −10−4 ⓐ (−4)−6ⓑ −4−6 ⓐ 14096ⓑ −14096 ⓐ 5·2−1ⓑ (5·2)−1 ⓐ 10·3−1ⓑ (10·3)−1 ⓐ 103ⓑ 130 ⓐ 4·10−3ⓑ (4·10)−3 ⓐ 3·5−2ⓑ (3·5)−2 ⓐ 325ⓑ 1225 n−4 p−3 1p3 c−10 m−5 1m5 ⓐ 4x−1ⓑ (4x)−1ⓒ (−4x)−1 ⓐ 3q−1ⓑ (3q)−1ⓒ (−3q)−1 ⓐ 3qⓑ 13qⓒ −13q ⓐ 6m−1ⓑ (6m)−1ⓒ (−6m)−1 ⓐ 10k−1ⓑ (10k)−1ⓒ (−10k)−1 ⓐ 10kⓑ 110kⓒ −110k Simplify Expressions with Integer Exponents In the following exercises, simplify. p−4·p8 r−2·r5 r³ n−10·n2 q−8·q3 1q5 k−3·k−2 z−6·z−2 1z8 a·a−4 m·m−2 1m p5·p−2·p−4 x4·x−2·x−3 1x a3b−3 u2v−2 u2v2 (x5y−1)(x−10y−3) (a3b−3)(a−5b−1) 1a2b4 (uv−2)(u−5v−4) (pq−4)(p−6q−3) 1p5q7 (−2r−3s9)(6r4s−5) (−3p−5q8)(7p2q−3) −21q5p3 (−6m−8n−5)(−9m4n2) (−8a−5b−4)(−4a2b3) 32a3b (a3)−3 (q10)−10 1q100 (n2)−1 (x4)−1 1x4 (y−5)4 (p−3)2 1p6 (q−5)−2 (m−2)−3 m⁶ (4y−3)2 (3q−5)2 9q10 (10p−2)−5 (2n−3)−6 n1864 u9u−2 b5b−3 b⁸ x−6x4 m5m−2 m⁷ q3q12 r6r9 1r3 n−4n−10 p−3p−6 p³ Convert from Decimal Notation to Scientific Notation In the following exercises, write each number in scientific notation. 45,000 280,000 2.8 × 10⁵ 8,750,000 1,290,000 1.29 × 10⁶ 0.036 0.041 4.1 × 10⁻² 0.00000924 0.0000103 1.03 × 10⁻⁵ The population of the United States on July 4, 2010 was almost 310,000,000. The population of the world on July 4, 2010 was more than 6,850,000,000. 6.85 × 10⁹ The average width of a human hair is 0.0018 centimeters. The probability of winning the 2010 Megamillions lottery is about 0.0000000057. 5.7 × 10⁻⁹ Convert Scientific Notation to Decimal Form In the following exercises, convert each number to decimal form. 4.1×102 8.3×102 830 5.5×108 1.6×1010 16,000,000,000 3.5×10−2 2.8×10−2 0.028 1.93×10−5 6.15×10−8 0.0000000615 In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was 2×104. At the start of 2012, the US federal budget had a deficit of more than $1.5×1013. $15,000,000,000,000 The concentration of carbon dioxide in the atmosphere is 3.9×10−4. The width of a proton is 1×10−5 of the width of an atom. 0.00001 Multiply and Divide Using Scientific Notation In the following exercises, multiply or divide and write your answer in decimal form. (2×105)(2×10−9) (3×102)(1×10−5) 0.003 (1.6×10−2)(5.2×10−6) (2.1×10−4)(3.5×10−2) 0.00000735 6×1043×10−2 8×1064×10−1 20,000,000 7×10−21×10−8 5×10−31×10−10 50,000,000

Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by 1.5 trillion calories by the end of 2015. ⓐ Write 1.5 trillion in decimal notation.ⓑ Write 1.5 trillion in scientific notation. Length of a year The difference between the calendar year and the astronomical year is 0.000125 day. ⓐ Write this number in scientific notation. ⓑ How many years does it take for the difference to become 1 day? ⓐ 1.25 × 10⁻⁴ⓐ 8,000 Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided 1 by 2,598,960 and saw the answer 3.848×10−7. Write the number in decimal notation. Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with 6 of her 50 favorite photographs, she multiplied 50·49·48·47·46·45. Her calculator gave the answer 1.1441304×1010. Write the number in decimal notation. 11,441,304,000

ⓐ Explain the meaning of the exponent in the expression 23. ⓑ Explain the meaning of the exponent in the expression 2−3 When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative? Answers will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

negative exponentIf n is a positive integer and a≠0, then a−n=1an. scientific notationA number expressed in scientific notation when it is of the form a×10n, where a≥1 and a<10, and n is an integer.