A.2 Exponents and Radicals - mrsk.ca

1 Appendix and RadicalsA11 Exponential NotationIf is a real number and is a positive integer, thenn factorswhere is the exponentandis the expression is read tothe th power. naananan a a a .. anaWhatyou should learn Use properties of Exponents . Use scientific notation to represent real numbers. Use properties of Radicals . Simplify and combine Radicals . Rationalize denominators andnumerators. Use properties of rational should learn itReal numbers and algebraicexpressions are often writtenwith Exponents and Radicals . Forinstance, in Exercise 105 on pageA22, you will use an expressioninvolving rational Exponents to find the time required for a funnel to empty for differentwater and ExponentsRepeated multiplicationcan be written in exponential MultiplicationExponential Form 2x 4 2x 2x 2x 2x 4 3 4 4 4 a5a a a a aAn exponent can also be negative.

2 In Property 3 below, be sure you see how touse a negative 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 nnmbaYo u c a n u s e a calculator to evalu-ate exponential expressions. Whendoing so, it is important to knowwhen to use parentheses becausethe calculator follows the order ofoperations. For instance, evaluateas followsScientific:24 Graphing:24 The display will be 16. If you omitthe parentheses, the display willbe 16.

3 2 4 Technologyyx > 12/6/05 2:12 PM Page A11A12 Appendix AReview of Fundamental Concepts of AlgebraIt is important to recognize the difference between expressions such asand In the parentheses indicate that the exponent applies tothe negative sign as well as to the 2, but in the exponent appliesonly to the 2. So,andThe properties of Exponents listed on the preceding page apply to allintegersand not just to positive integers as shown in the examples in this properties of ExponentsUse the properties of Exponents to simplify each try Exercise with Positive ExponentsRewrite each expression with positive exponent does not apply to 3 and 5 and 7 Property 6 Property try Exercise 33. 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 5x3x 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 n,m 24 16.

4 2 4 16 24 24 , 2 4, 24. 2 4 Rarely 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. 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 Example2(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 Example 1 Example 12/6/05 2:12 PM Page A12 Appendix and RadicalsA13 Most 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 expo-nential entry key the user s guide for your calculator for instructions on keystrokes and hownumbers in scientific notation are NotationExponents provide an efficient way of writing and computing with very large (orvery small) numbers.

5 For instance, there are about 359 billion billion gallons ofwater on Earth that is, 359 followed by 18 ,000,000,000,000,000,000It is convenient to write such numbers in scientific notation. This notation hasthe form where and is an integer. So, the number ofgallons of water on Earth 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 thatthe decimal point has been moved 20 places. A negativeexponent indicates thatthe number is small(less than 1). For instance, the mass (in grams) of one elec-tron is decimal placesScientific NotationWrite each number in scientific ,100, try Exercise NotationWrite each number in decimal try Exercise 102 10 6 10 6836,100,000 10 10 28 1020.

6 N1 c<10 c 10n,TechnologyExample 3 Example 12/6/05 2:12 PM Page A13A14 Appendix AReview of Fundamental Concepts of AlgebraRadicals and Their PropertiesA square rootof a number is one of its two equal factors. For example, 5 is asquare root 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 aresquare roots of 25. The principal square rootof 25, written as is the positiveroot, 5. The principal nth rootof a number is defined as common misunderstanding is that the square root sign implies bothnegative and positive roots. This is not correct. The square root sign implies onlya positive root.

7 When a negative root is needed, you must use the negative signwith the square root :Correct:andEvaluating Expressions Involving not a real number because there is no real number that can be raisedto the fourth power to produce Now try Exercise 51. 81 2 5 32 2 54 3 5343 12564 .3 12564 54 36 62 6 6. 36 662 36. 36 6 4 2 4 2 4 2 25, 5125 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 theroot 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. The principal nth rootof ais the nth root that has the same sign as a. It is denoted by a radicalsymbolPrincipal nth rootThe positive integer nis the indexof the radical, and the number ais omit the index and write rather than (Theplural of index is indices.)

8 2 a. an 2,n 12/6/05 2:12 PM Page A14 Appendix and RadicalsA15 Here are some generalizations about the nth roots of real such as 1, 4, 9, 16, 25, and 36 are called perfect squaresbecausethey have integer square roots. Similarly, integers such as 1, 8, 27, 64, and 125are called perfect 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 a2 a . properties of RadicalsLet aand bbe real numbers, variables, or algebraic expressions such thatthe indicated roots are real numbers, and let mand nbe positive neven,For nodd,3 12 3 12n an a. 12 2 12 12n an a.

9 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 mReal NumberaIntegernRoot(s) ofaExampleis is real rootsis not a real even or 0 0n 0 0na 0 4na<03 8 2n ana<0a>0 4 81 34 81 3, n an a,n>0,a>0 Generalizations About nth Roots of Real NumbersExample 12/6/05 2:12 PM Page A15A16 Appendix AReview of Fundamental Concepts of AlgebraWhen you simplify a radical, itis important that both expres-sions are defined for the samevalues of the variable. Forinstance, in Example 7(b),and are bothdefined only for nonnegativevalues of Similarly, inExample 7(c),and are both defined for all realvalues of x 4 5x 3x 75x3 Simplifying RadicalsAn expression involving Radicals is in simplest formwhen the followingconditions are possible factors have been removed from the fractions have radical-free denominators (accomplished by a processcalled rationalizing the denominator).

10 Index of the radical is reduced. To simplify a radical, factor the radicand into factors whose Exponents aremultiples of the index. The roots of these factors are written outside the radical,and the leftover factors make up the new Even RootsPerfectLeftover4th largest square root of perfect try Exercise 63(a).Simplifying Odd largest cube root of perfect largest cube root of perfect try Exercise 63(b). 2x23 5 3 2x2 3 53 40x6 3 8x6 5 2a3 3a 3 2a 3 3a3 24a4 3 8a3 3a3 24 3 8 3 3 23 3 23 34 5x 4 5x 5 x 5x 3x 5x 2 3x 75x3 25x2 3x4 48 4 16 3 4 24 3 24 3 Example 7 Example 12/6/05 2:12 PM Page A16 Appendix and RadicalsA17 Radical expressions can be combined (added or subtracted) if they are likeradicals that is, if they have the same index and radicand.