594Chapter 9 <strong>Exponents</strong> <strong>and</strong> <strong>Polynomials</strong>35. 10 −2 ÷ 10 536. 10 −4 ÷ 10 237. 10 −5 ÷ 10 238. 10 −3 ÷ 10_110 7 _ 110 6 _ 110 7 _ 110 439. _10−210 −51,00040. _10−410 −71,00041. x 12 ÷ x −12x 2442. x 6 ÷ x −6x 1243. 2 12 ÷ 2 10444. 3 11 ÷ 3 9945. 10 _10 −310,00046. 102 _10 −510 7A B Simplify each expression. Write your answer using only positive exponents.47. 3x −2 ⋅ 5x 748. 5x −3 ⋅ 8x 615x 5 40x 349. 2x −2 ⋅ 3x −3 6 _x 550. 4x −6 ⋅ 5x −4_ 20x 1051. (4x −4 )(−3x −3 )52. (3x 2 )(−4x −1 )_− 12x −12x753. 7x ⋅ 3x −4 ⋅ 2x 554. 6x ⋅ 2x −3 ⋅ 3x 442x 2 36x 255. _x −3 ⋅ x 7_x −256. x −4 ⋅ x 6x −1x 5 x 457. _x −2 ⋅ x −8x 2x −1258. _x −5 ⋅ x −7x 3x −15Maintaining Your SkillsMake the following conversions.59. 6 minutes 45 seconds toa. Seconds405 secb. Minutes6.75 min60. 4 minutes 15 seconds toa. Seconds255 secb. Minutes4.25 min61. 3 pounds 8 ounces toa. Ounces56 ozb. Pounds3.5 lb62. 5 pounds 4 ounces toa. Ounces84 ozb. Pounds5.25 lb63. 5 feet 9 inches toa. Inches69 in.b. Feet5.75 ft64. 6 feet 3 inches toa. Inches75 in.b. Feet6.25 ft
Scientific NotationThere are many disciplines that deal with very large numbers <strong>and</strong> others that dealwith very small numbers. For example, in astronomy, distances commonly aregiven in light-years. A light-year is the distance that light will travel in one year. Itis approximately9.5ObjectivesA Write numbers in scientific notation.B Convert numbers written inscientific notation to st<strong>and</strong>ard form.5,880,000,000,000 milesIt can be difficult to perform calculations with numbers in this form because of thenumber of zeros present. Scientific notation provides a way of writing very large,or very small, numbers in a more manageable form.A Scientific NotationExamples now playing atMathTV.com/booksDefinitionA number is in scientific notation when it is written as the product of anumber between 1 <strong>and</strong> 10 <strong>and</strong> an integer power of 10. A number written inscientific notation has the formn × 10 rwhere 1 ≤ n < 10 <strong>and</strong> r = an integer.Example 1The speed of light is 186,000 miles per second. Write186,000 in scientific notation.Solution To write this number in scientific notation, we rewrite it as the productof a number between 1 <strong>and</strong> 10 <strong>and</strong> a power of 10. To do so, we move the decimalpoint 5 places to the left so that it appears between the 1 <strong>and</strong> the 8, giving us 1.86.Then we multiply this number by 10 5 . The number that results has the same valueas our original number but is written in scientific notation. Here is our result:Practice Problems1. Write 27,500 in scientificnotation.186,000 = 1.86 × 10 5Both numbers have exactly the same value. The number on the left is written inst<strong>and</strong>ard form, while the number on the right is written in scientific notation.BConverting to St<strong>and</strong>ard FormExample 2If your pulse rate is 60 beats per minute, then your heartwill beat 8.64 × 10 4 times each day. Write 8.64 × 10 4 in st<strong>and</strong>ard form.Solution Because 10 4 is 10,000, we can think of this as simply a multiplicationproblem. That is,2. Write 7.89 × 10 5 in st<strong>and</strong>ardform.8.64 × 10 4 = 8.64 × 10,000 = 86,400Looking over our result, we can think of the exponent 4 as indicating the numberof places we need to move the decimal point to write our number in st<strong>and</strong>ardform. Because our exponent is positive 4, we move the decimal point from itsoriginal position, between the 8 <strong>and</strong> the 6, four places to the right. If we need toadd any zeros on the right we do so. The result is the st<strong>and</strong>ard form of our number,86,400.9.5 Scientific NotationAnswers1. 2.75 × 10 4 2. 789,000595