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Simplifying Square Roots of Square Roots by Denesting 63 4 Simplifying Square Ro ots of Square Ro ots by Denesting David J. Je rey Alb ert D. Rich The UniversityofWestern Ontario Soft Warehouse, Inc. Abstract: We discuss why it is imp ortant to try to simplify the square ro ot of an expression containing other square ro ots, and we give rules for doing this when it is p ossible. The square ro ot of an expression containing nth ro ots is also considered brie y. This article, in addition to treating a sp eci c mathematical topic, shows some of the steps that develop ers must take when writing computer algebra systems. 4.1 Intro duction p p 3 1 2 and 5 are called surds [Chrystal64, Hall88]. The more general Numb ers suchas terms radical and algebraic number are also used, but surd is the most sp eci c. It is 2 avenerable term that has b een revived for use in computer systems . Given p ositive n integers n and k , the unique p ositive real ro ot of the equation x = k will b e denoted p n k and called a surd. Most b o oks [Chrystal64, Hall88]allow k to b e a p ositive rational numb er, and there is no great di erence in principle if we accept that generalization; in this article, however, all the examples use integers. Another p ointofvariation is p p 8or2 2; here we the treatment of p erfect-p ower factors, i.e., whether one writes use whichever form is more convenient. The term radical may b e familiar to some readers from the common phrase `solvable in terms of radicals', which is used whenever a textb o ok discusses the ro ots of a p olynomial [Dickson26]. A p olynomial equation is solvable by radicals if `. its ro ots can b e found by rational op erations and extractions of a ro ot. ' [Dickson26]. Thus p p a quantity suchas 1+ 2 is a radical, but not a surd. Polynomials of degree less 1 Please do not tell us jokes ab out these numb ers b eing ab surd. Wehave heard them all b efore. p 2 The term surd is used byT X as the name for the symbol ; Maple has a function called E surd that is similar to the nth ro ot de ned here; like all go o d mathematical terms, the precise de nition dep ends up on the context. In general, a mathematical term that do es not have several con icting de nitions is not imp ortant enough to b e worth learning. From Computer Algebra Systems: A Practical Guide, M.J. Wester, Ed., Wiley 1999. 62 COMPUTER ALGEBRA SYSTEMS: A PRACTICAL GUIDE than or equal to 4 are always solvable by radicals, and higher-degree p olynomials are sometimes solvable by radicals. An algebraic number is a ro ot of some p olynomial with integer co ecients, and in general will not b e expressible as a radical, although all radicals are sp ecial cases of algebraic numb ers. The p ositive real solution of the n equation x = r , where r is a p ositive radical, will here b e called the nth ro ot of r , and will itself b e a radical. The other familiar name for these typ es of numb ers is (1=n)th power, but this term will not b e used b ecause fractional p owers are regarded bymany 3 p eople as multivalued functions , and wewanttowork with uniquely de ned (and named) quantities. Our attention in this article is directed to radicals consisting of the square ro ot of an p p p expression containing surds, for example 2+ 3 . Such expressions are often called nestedradicals, although one could argue that strictly the word nested is redundant. Such expressions can sometimes b e simpli ed, but the rules for such simpli cations are not discussed in standard b o oks. Here are some examples of simpli cations. The rst example shows that two square ro ots can sometimes b e reduced to one: q p p 3+2 2=1+ 2 : (4.1) Sometimes the numb er of square ro ot op erations remains unchanged, but p eople prefer the format of the new expression: q p p p 5+2 6= 2+ 3 : (4.2) This rearrangement of the square-ro ot op erations is usually called denesting. Here are some more examples of square-ro ot denesting. q p p p p 4 4 5 3+ 6 2 = 27 + 12 ; (4.3) q p p p p p p 12 + 2 6+ 2 14 + 2 21 = 2+ 3+ 7 : (4.4) We do not restrict ourselves to taking the square ro ot only of other square ro ots: q p p p 3 3 3 9+ 6 3+9 = 3+ 3 ; (4.5) q p p p p p 3 3 3 3 3 1 5 4 = : (4.6) 2+ 20 25 3 4.2 Why denest square ro ots? Discovering the relations given in the examples ab oveisaninteresting mathematical challenge, but why should a computer algebra system (CAS) devote computing resources to denesting problems? The rst reason is simplicity. Users of CAS, likeall mathematicians, prefer to see their results expressed in `simplest' form. Most p eople would regard the right-hand sides of our examples ab ove as b eing `simpler' than the left-hand sides, although there mightbesomewhowould disagree. For example, in 3 The question \Are fractional p owers multivalued?" is a standard basis for table assignments at conference banquets. From Computer Algebra Systems: A Practical Guide, M.J. Wester, Ed., Wiley 1999. SIMPLIFYING SQUARE ROOTS OF SQUARE ROOTS BY DENESTING 63 [Landau92b], reasons are given for preferring the left-hand side of (4.2). However, we 4 assume that in general users would want denesting to b e discovered . Another reason for denesting is reliable simpli cation.For b oth p eople and computers, there is a danger that the result of a mathematical simpli cation will dep end up on the order in which rules are applied. For example, the simpli cation q p 2 of (1 2) can pro ceed twoways. The rst way is called `top-down' bythose who think of the expression as a tree, and `outside to middle' by those who lo ok p 2 x = jxj at the printed form. Following this pro cedure, one applies rst the rule q p p 2 for any real x, and obtains (1 2) = 2 1. The other way is `b ottom-up' or p p `middle outwards', in which one expands the square and obtains 3 2 2. Without denesting, the simpli cation will not pro ceed any further. Thus two p eople using the same CAS to solve the same problem could arrive at apparently di erent answers, and 5 this is what we wish to avoid . A nal reason is a small improvement in the accuracy of the numerical evaluation of some radical expressions. For example, the quantity q p 199999 600 111110 0:00158 approximates to zero if 10 decimal digits of precision are used, and only at 15 digits p p 10 3 11111 is the expression found to b e nonzero; in contrast, the equivalent 100 approximates to 0:00158 using just 7 digits of precision. 4.3 Where do nested radicals arise? Older algebra textb o oks contain explicit problems on nested radicals, for example, the b o oks by Chrystal [Chrystal64] and Hall & Knight [Hall88], which are b oth 19th century b o oks (the 1964 date on the Chrystal citation is not indicative of the age of the b o ok, whose rst edition was published in 1886). Clearly, users could challenge a computer system with problems from such b o oks, but there are other sources of problems. Even if users do not delib erately p ose denesting problems, the problems they do p ose can generate subproblems that contain nested radicals. Many problems in mathematics have solutions expressed as standard formulae. For example, all students learn quite early in their studies the formula for solving a quadratic equation; not surprisingly, it is programmed into most CAS. Consider what happ ens when that formula is used for the problem p 2 x +6x 4 5= 0 : The standard formula gives q p 9+4 5 ; x = 3 4 Long after ordinary text was set mechanically, mathematical equations were still set by hand. Long horizontal lines, as in large fractions or large ro ots, were troublesome for the comp ositors, and authors of mathematicswere encouraged to prefer forms that reduced the need for such lines. Perhaps nowthatwehaveT X to set these expressions easily,our E mathematical tastes with resp ect to simplicity will change also. 5 Not to mention all the phone calls to technical supp ort. From Computer Algebra Systems: A Practical Guide, M.J. Wester, Ed., Wiley 1999. 64 COMPUTER ALGEBRA SYSTEMS: A PRACTICAL GUIDE p p but in fact the quadratic can b e factored as (x +1 5)(x +5+ 5). Therefore the general formula is correct, but do es not directly give the simplest result in this sp ecial case. The problem is an interesting one to try on humans, as well as computer systems. 2 Now consider the formula, valid for real a; b with b>a , Z 2x + a 1 1 p p arctan : (4.7) dx = 2 2 2 4x +4ax + b 2 b a b a p If we substitute a =2 and b = 7+2 2, we nd that (4.7) b ecomes Z 1 1 2x +2 p p p dx = arctan : p p 2 4x +8x + 7+2 2 2 3+2 2 3+ 2 2 As example (4.1) shows, however, the nested radicals on the right-hand side can b e p 2 b a must remain simpli ed.
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