Exponents and 10 Scientifi c Notation 110.10.1 EExponentsxponents 110.20.2 PProductroduct ooff PPowersowers PPropertyroperty 110.30.3 QQuotientuotient ooff PPowersowers PPropertyroperty 110.40.4 ZZeroero aandnd NNegativeegative EExponentsxponents 110.50.5 RReadingeading SScientificientifi c NNotationotation 110.60.6 WWritingriting SScientificientifi c NNotationotation 10.7 Operations in Scientifi c Notation are of my friends enemies of my are my “The . The enemies friends enemies friends.” friends of my my are my “The friends of my enemies friends. The .” are my enemies “Here’s how it goes, Descartes.” enemiese “If one flea had 100 babies, and each baby grew up and had 100 babies, ...” “... and each of those babies grew up and had 100 babies, you would have 1,010,101 fleas.” What You Learned Before “It’s called the 2 Negative One Power of Example 1 Evaluate 6 ÷ 4 − 2(9 − 5). , Descartes!” First: Parentheses 62 ÷ 4 − 2(9 − 5) = 62 ÷ 4 − 2 ⋅ 4 Second: Exponents = 36 ÷ 4 − 2 ⋅ 4 Third: Multiplication and Division (from left to right) = 9 − 8 Fourth: Addition and Subtraction (from left to right) = 1 Evaluate the expression. 8 1. 15 — + 22 − 3 ⋅ 7 2. 52 ⋅ 2 ÷ 10 + 3 ⋅ 2 − 1 3. 32 − 1 + 2(4(3 + 2)) ( 4) Example 2 Find 2.1 ⋅ 0.35. Example 3 Find 1.08 ÷ 0.9. ) 2.1 1 decimal place 0.9 ‾1.08 Multiply each number by 10. × 0.3 5 + 2 decimal places 1 0 5 6 3 1.2 Place the decimal point ) 0.7 3 5 3 decimal places 9 ‾10.8 above the decimal point − 9 in the dividend 10.8. 1 8 − 1 8 0 Find the product or quotient. 4. 1.75 ⋅ 0.2 5. 1.4 ⋅ 0.6 6. 7.03 7. 0.894 × 4.3 × 0.2 8. 5.40 ÷ 0.09 9. 4.17 ÷ 0.3 10. 0.15 )‾ 3.6 11. 0.004 )‾ 7.2 10.1 Exponents How can you use exponents to write numbers? The expression 35 is called a power. The base is 3. The exponent is 5. base 35 exponent 1 ACTIVITY: Using Exponent Notation Work with a partner. a. Copy and complete the table. Power Repeated Multiplication Form Value (−3)1 −3 −3 (−3)2 (−3) ⋅ (−3) 9 (−3)3 (−3)4 (−3)5 (−3)6 (−3)7 b. REPEATED REASONING Describe what is meant by the expression (−3)n. How can you fi nd the value of (−3)n ? 2 ACTIVITY: Using Exponent Notationatition COMMON Work with a partner. CORE a. The cube at the right has $3 in each Exponents In this lesson, you will of its small cubes. Write a power ● write expressions using that represents the total amount integer exponents. of money in the large cube. ● evaluate expressions involving integer b. Evaluate the power to fi nd the exponents. total amount of money in the Learning Standard large cube. 8.EE.1 410 Chapter 10 Exponents and Scientifi c Notation 3 ACTIVITY: Writing Powers as Whole Numbers Work with a partner. Write each distance as a whole number. Which numbers do you know how to write in words? For instance, in words, 103 is equal to one thousand. a. 1026 meters: b. 1021 meters: diameter of diameter of observable Milky Way galaxy universe c. 1016 meters: d. 107 meters: diameter of diameter of solar system Earth e. 106 meters: f. 105 meters: Width length of width of Lake Erie Lake Erie shoreline 4 ACTIVITY: Writing a Power Math Work with a partner. Write the number Practice of kits, cats, sacks, and wives as a power. Analyze Givens As I was going to St. Ives What information is I met a man with seven wives given in the poem? Each wife had seven sacks What are you trying Each sack had seven cats to fi nd? Each cat had seven kits Kits, cats, sacks, wives How many were going to St. Ives? Nursery Rhyme, 1730 5. IN YOUR OWN WORDS How can you use exponents to write numbers? Give some examples of how exponents are used in real life. Use what you learned about exponents to complete Exercises 3–5 on page 414. Section 10.1 Exponents 411 10.1 Lesson Lesson Tutorials A power is a product of repeated factors. The base of a power is the Key Vocabulary common factor. The exponent of a power indicates the number of times power, p. 412 the base is used as a factor. base, p. 412 exponent, p. 412 base exponent 1 5 1 1 1 1 1 — = — ⋅ — ⋅ — ⋅ — ⋅ — ( 2) 2 2 2 2 2 1 power — is used as a factor 5 times. 2 EXAMPLE 1 Writing Expressions Using Exponents Write each product using exponents. Study Tip a. (−7) ⋅ (−7) ⋅ (−7) − Use parentheses to Because 7 is used as a factor 3 times, its exponent is 3. write powers with negative bases. So, (−7) ⋅ (−7) ⋅ (−7) = (−7)3. b. π ⋅ π ⋅ r ⋅ r ⋅ r Because π is used as a factor 2 times, its exponent is 2. Because r is used as a factor 3 times, its exponent is 3. So, π ⋅ π ⋅ r ⋅ r ⋅ r = π 2r 3. Write the product using exponents. 1 1 1 1 1 Exercises 3–10 1. — ⋅ — ⋅ — ⋅ — ⋅ — 2. 0.3 ⋅ 0.3 ⋅ 0.3 ⋅ 0.3 ⋅ x ⋅ x 4 4 4 4 4 EXAMPLE 2 Evaluating Expressions Evaluate each expression. a. (−2)4 (−2)4 = (−2) ⋅ (−2) ⋅ (−2) ⋅ (−2) Write as repeated multiplication. The base is −2. = 16 Simplify. b. −24 −24 = −(2 ⋅ 2 ⋅ 2 ⋅ 2) Write as repeated multiplication. The base is 2. = −16 Simplify. 412 Chapter 10 Exponents and Scientifi c Notation EXAMPLE 3 Using Order of Operations Evaluate each expression. a. 3 + 2 ⋅ 34 = 3 + 2 ⋅ 81 Evaluate the power. = 3 + 162 Multiply. = 165 Add. b. 33 − 82 ÷ 2 = 27 − 64 ÷ 2 Evaluate the powers. = 27 − 32 Divide. = −5 Subtract. Evaluate the expression. Exercises 11–16 1 3 3. −54 4. − — 5. ∣ −33 ÷ 27 ∣ 6. 9 − 25 ⋅ 0.5 and 21–26 ( 6) EXAMPLE 4 Real-Life Application In sphering, a person is secured inside a small, hollow sphere that is surrounded by a larger sphere. The space between the spheres is infl ated with air. What is the volume of the infl ated space? You can fi nd the radius of each sphere by dividing each diameter given in the diagram by 2. Outer Sphere Inner Sphere 4 4 V = — π r 3 Write formula. V = — π r 3 2 m 3 m 3 3 4 3 3 4 = — π — Substitute. = — π (1)3 3 ( 2) 3 4 27 4 = — π — Evaluate the power. = — π (1) 3 ( 8 ) 3 9 4 = — π Multiply. = — π 2 3 9 4 19 So, the volume of the infl ated space is — π − — π = — π, 2 3 6 or about 10 cubic meters. 7. WHAT IF? The diameter of the inner sphere is 1.8 meters. What is the volume of the infl ated space? Section 10.1 Exponents 413 10.1 Exercises Help with Homework 1. NUMBER SENSE Describe the difference between − 34 and (− 3)4. 2. WHICH ONE DOESN’T BELONG? Which one does not belong with the other three? Explain your reasoning. 53 53 53 53 The exponent is 3. The power is 5. The base is 5. Five is used as a factor 3 times. 6)=3 9+(- )= +(-3 3 9)= 4+(- 1)= 9+(- Write the product using exponents. 1 3. 3 ⋅ 3 ⋅ 3 ⋅ 3 4. (−6) ⋅ (−6) 1 1 1 1 1 1 5. − — ⋅ − — ⋅ − — 6. — ⋅ — ⋅ — ( 2) ( 2) ( 2) 3 3 3 7. π ⋅ π ⋅ π ⋅ x ⋅ x ⋅ x ⋅ x 8. (−4) ⋅ (−4) ⋅ (−4) ⋅ y ⋅ y 9. 6.4 ⋅ 6.4 ⋅ 6.4 ⋅ 6.4 ⋅ b ⋅ b ⋅ b 10. (−t) ⋅ (−t) ⋅ (−t) ⋅ (−t) ⋅ (−t) Evaluate the expression. 2 11. 52 12. −113 13. (−1)6 1 6 1 2 1 3 14. — 15. − — 16. − — ( 2) ( 12) ( 9) 17. ERROR ANALYSIS Describe and correct the error in evaluating the expression. ✗ −62 = (−6) ⋅ (−6) = 36 18. PRIME FACTORIZATION Write the prime factorization of 675 using exponents. 1 1 1 1 19. STRUCTURE Write − — ⋅ — ⋅ — ⋅ — using exponents. ( 4 4 4 4) 20. RUSSIAN DOLLS The largest doll is 12 inches 7 tall. The height of each of the other dolls is — 10 the height of the next larger doll. Write an expression involving a power for the height of the smallest doll. What is the height of the smallest doll? 414 Chapter 10 Exponents and Scientifi c Notation Evaluate the expression. 3 2 2 2 3 21. 5 + 3 ⋅ 2 22. 2 + 7 ⋅ (−3) 23. ( 13 − 12 ) ÷ 5 1 1 1 3 1 2 24. — ( 43 − 6 ⋅ 32 ) 25. ∣ — ( 7 + 53 ) ∣ 26. ∣ − — ÷ — ∣ 2 2 ( 2) ( 4) 27. MONEY You have a part-time job. One day your boss offers to pay you either − 2 h − 1 or 2h 1 dollars for each hour h you work that day. Copy and complete the table. Which option should you choose? Explain. h 12345 2 h − 1 2 h − 1 28. CARBON-14 DATING Scientists use carbon-14 dating to determine the age of a sample of organic material. a. The amount C (in grams) of a 100-gram sample of carbon-14 remaining after t years is represented by the equation C = 100(0.99988)t. Use a calculator to fi nd the amount of carbon-14 remaining after 4 years. b. What percent of the carbon-14 remains after 4 years? 29. The frequency (in vibrations per second) of a note on a piano is represented by the equation F = 440(1.0595)n, where n is the number of notes above A-440.