A.2 EXPONENTS AND RADICALS - Cengage

1 Integer ExponentsRepeated multiplicationcan be written in exponential MultiplicationExponential Form 2x 4 2x 2x 2x 2x 4 3 4 4 4 a5a a a a aA14 Appendix AReview of Fundamental Concepts of AND RADICALSWhat you should learn Use properties of EXPONENTS . Use scientific notation to representreal numbers. Use properties of RADICALS . Simplify and combine RADICALS . Rationalize denominators andnumerators. Use properties of rational you should learn itReal numbers and algebraic expressionsare often written with EXPONENTS andradicals. For instance, in Exercise 121on page A26, you will use an expressioninvolving rational EXPONENTS to find thetimes required for a funnel to empty fordifferent water NotationIf is a real number and is a positive integer, thenn factorswhere is the exponentandis the expression is read to the thpower.

2 Naananan a a a .. anaProperties of ExponentsLet and be real numbers, variables, or algebraic expressions, and let and be integers. (All denominators and bases are nonzero.) 2 2 2 2 2 2 4 a2 a 2 a2 2x 3 23x3 8x3 ab m ambm y3 4 y3( 4) y 12 1y12 am n amn 5x 3 53x3 125x3 ab m ambm x2 1 0 1a 0a0 1,y 4 1y4 1y 4a n 1an 1a nx7x4 x7 4 x3aman am n32 34 32 4 36 729aman am nnmbaAn exponent can also be negative. In Property 3 below, be sure you see how to use anegative can use a calculator to evaluate exponential doing so, it is important toknow when to use parenthesesbecause the calculator follows theorder of operations. For instance,evaluate as :24 Graphing:24 The display will be 16.

3 If youomit the parentheses, the displaywill be 16. 2 4 ENTER > yxAppendix and RadicalsA15It is important to recognize the difference between expressions such as andIn the parentheses indicate that the exponent applies to the negative signas well as to the 2, but in the exponent applies only to the 2. So,andThe properties of EXPONENTS listed on the preceding page apply to allintegers andnot just to positive integers, as shown in the examples in this Exponential sign is part of the sign is notpart of the 2 and 3 Now try Exercise Algebraic ExpressionsEvaluate each algebraic expression when the expression has a value the expression has a value ofNow try Exercise properties of ExponentsUse the properties of EXPONENTS to simplify each try Exercise 31.

4 5x3y 2 52 x3 2y2 25x6y2a 03a 4a2 0 3a 1 3a, 2xy2 3 23 x 3 y2 3 8x3y6 3ab4 4ab 3 3 4 a a b4 b 3 12a2b 5x3y 23a 4a2 0 2xy2 3 3ab4 4ab 3 Example 313 x 3 13 3 3 13 27 x 3x 3,5x 2 5 3 2 532 2x 3,13 x 35x 2x 24446 44 6 4 2 142 1162 24 21 4 25 32 52 5 5 25 5 2 5 5 25 Example 1n,m 24 16. 2 4 16 24 24 , 2 4, 24. 2 4 Rewriting with Positive ExponentsRewrite each expression with positive exponent does not apply to 3 Property 5 and 7 Property 6 Property try Exercise NotationExponents provide an efficient way of writing and computing with very large (or verysmall) numbers. For instance, there are about 359 billion billion gallons of water onEarth that is, 359 followed by 18 ,000,000,000,000,000,000It is convenient to write such numbers in scientific notation.

5 This notation has the formwhere and is an integer. So, the number of gallons of water onEarth can be written in scientific notation ,000,000,000,000,000,000 The positiveexponent 20 indicates that the number is large(10 or more) and that the decimal point has been moved 20 places. A negativeexponent indicates that thenumber is small(less than 1). For instance, the mass (in grams) of one electron decimal 10 28 1020. n1 c<10 c 10n, y29x4 y232x4 3 2x 4y 2 3x2y 2 3 2 x2 2y 2 3a5b512a3b 44a 2b 12a3 a24b b4 213x 2 1 x2 3 x23x 1 1x 3x2y 212a3b 44a 2b13x 2x 1 Example 4A16 Appendix AReview of Fundamental Concepts of AlgebraRarely in algebra is there onlyone way to solve a t be concerned if the stepsyou use to solve a problem arenot exactly the same as the stepspresented in this text.

6 Theimportant thing is to use stepsthat you understand and, ofcourse, steps that are justifiedby the rules of algebra. Forinstance, you might prefer thefollowing steps for Example 4(d).Note how Property 3 is used in the first step of this fractional form of thisproperty is ab m ba m. 3x2y 2 y3x2 2 y29x4 Appendix and RadicalsA17 Scientific NotationWrite each number in scientific ,100, try Exercise NotationWrite each number in decimal try Exercise 102 10 6 102 10 6 Example 6836,100,000 10 5 Example 5 TECHNOLOGYMost calculators automatically switch to scientific notation when they are showinglarge (or small) numbers that exceed the display enternumbers in scientific notation, your calculator should have an exponentialentry key the user s guide for your calculator for instructions on keystrokes and hownumbers in scientific notation are Scientific NotationEvaluate SolutionBegin by rewriting each number in scientific notation and try Exercise 63(b).

7 240,000 105 103 10 2 2,400,000,000 1500 109 10 6 10 5 103 2,400,000,000 1500 .Example 7 EEEXPR adicals and Their PropertiesA square rootof a number is one of its two equal factors. For example, 5 is a squareroot of 25 because 5 is one of the two equal factors of 25. In a similar way, a cube rootof a number is one of its three equal factors, as in Some numbers have more than one nth root. For example, both 5 and are squareroots of 25. The principal square rootof 25, written as is the positive root, 5. Theprincipal nth rootof a number is defined as common misunderstanding is that the square root sign implies both negative andpositive roots.

8 This is not correct. The square root sign implies only a positive a negative root is needed, you must use the negative sign with the square :Correct:andEvaluating Expressions Involving not a real number because there is no real number that can be raised to thefourth power to produce Now try Exercise 65. 81 2 5 32 2 54 3 5343 12564 .3 12564 54 36 62 6 6. 36 662 36. 36 6 Example 8 4 2 4 2 4 2 25, 5125 AReview of Fundamental Concepts of AlgebraDefinition of nth Root of a NumberLet aand bbe real numbers and let be a positive integer. Ifthen bis an nth root of the root is a square the root is a cube 3,n 2,a bnn 2 Principal nth Root of a NumberLet abe a real number that has at least one nth root.

9 The principal nth root of ais the nth root that has the same sign as a. It is denoted by a radical symbolPrincipal nth rootThe positive integer nis the indexof the radical , and the number ais the omit the index and write rather than (The plural of index isindices.)2 a. an 2,n and RadicalsA19 Here are some generalizations about the nth roots of real such as 1, 4, 9, 16, 25, and 36 are called perfect squaresbecause theyhave integer square roots. Similarly, integers such as 1, 8, 27, 64, and 125 are calledperfect cubesbecause they have integer cube common special case of Property 6 is Using properties of RadicalsUse the properties of RADICALS to simplify each try Exercise y6 y 3 x3 x 3 5 3 5 8 2 8 2 16 46 y63 x3 3 5 3 8 2 Example 9 a2 a.

10 properties of RadicalsLet aand bbe real numbers, variables, or algebraic expressions such that theindicated roots are real numbers, and let mand nbe positive neven,For nodd,3 12 3 12n an a. 12 2 12 12n an a . 3 2 3 n a n a3 10 6 10m n a mn a4 274 9 4 279 4 3b 0n an b n ab , 5 7 5 7 35n a n b n ab3 82 3 8 2 2 2 4n am n a mGeneralizations About nth Roots of Real NumbersReal NumberaIntegernRoot(s) ofaExamplea>0is >0, n an a, 4 81 34 81 3,or a<0a>0is a3 8 2a<0is real rootsis not a real number. 4a 0is even or 0 05 0 0 WARNING / CAUTIONWhen you simplify a radical , itis important that both expressionsare defined for the same valuesof the variable.