Nth rootFrom Wikipedia, the free encyclopediaContents1 Nested intervals 11.1 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Nested radical 32.1 Denesting nested radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Some identities of Ramanujan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Landaus algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 In trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 In the solution of the cubic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Innitely nested radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.1 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 nth root 93.1 Etymology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.1 Origin of the root symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Etymology of surd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Denition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.2 Cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Identities and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Simplied form of a radical expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 Computing principal roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7.1 nth root algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7.2 Digit-by-digit calculation of principal roots of decimal (base 10) numbers . . . . . . . . . 173.7.3 Logarithmic computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18iii CONTENTS3.8 Geometric constructibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 Complex roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.9.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.9.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.9.3 nth roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.10 Solving polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.14 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 233.14.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.14.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.14.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Chapter 1Nested intervals0000In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbersInsuch that each set In is an interval of the real line, for n = 1, 2, 3, ..., and that furtherIn is a subset of Infor all n. In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left.The main question to be posed is the nature of the intersection of all the In. Without any further information, all thatcan be said is that the intersection J of all the In, i.e. the set of all points common to the intervals, is either the emptyset, a point, or some interval.The possibility of an empty intersection can be illustrated by the intersection when In is the open interval(0, 2n).Here the intersection is empty, because no number x is both greater than 0 and less than every fraction 2n.The situation is dierent for closed intervals. The nested intervals theoremstates that if each In is a closed and boundedinterval, sayIn = [an, bn]with12 CHAPTER 1. NESTED INTERVALSan bnthen under the assumption of nesting, the intersection of the In is not empty. It may be a singleton set {c}, or anotherclosed interval [a, b]. More explicitly, the requirement of nesting means thatan an andbn bn .Moreover, if the length of the intervals converges to 0, then the intersection of the In is a singleton.One can consider the complement of each interval, written as(, an) (bn, ) . By De Morgans laws, thecomplement of the intersection is a union of two disjoint open sets. By the connectedness of the real line there mustbe something between them. This shows that the intersection of (even an uncountable number of) nested, closed, andbounded intervals is nonempty.1.1 Higher dimensionsIn two dimensions there is a similar result: nested closed disks in the plane must have a common intersection. Thisresult was shown by Hermann Weyl to classify the singular behaviour of certain dierential equations.1.2 See alsoBisectionCantors Intersection Theorem1.3 ReferencesFridy, J. A. (2000), 3.3 The Nested Intervals Theorem, Introductory Analysis: The Theory of Calculus,Academic Press, p. 29, ISBN 9780122676550.Shilov, Georgi E. (2012), 1.8 The Principle of Nested Intervals, Elementary Real and Complex Analysis,Dover Books on Mathematics, Courier Dover Publications, pp. 2122, ISBN 9780486135007.Sohrab, Houshang H. (2003), Theorem 2.1.5 (Nested Intervals Theorem)", Basic Real Analysis, Springer, p.45, ISBN 9780817642112.Chapter 2Nested radicalIn algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) thatcontains (nests) another radical expression. Examples include:5 25which arises in discussing the regular pentagon;5 + 26 ,or more complicated ones such as:32 +3 +34 .2.1 Denesting nested radicalsSome nested radicals can be rewritten in a form that is not nested. For example,3 + 22 = 1 +2 ,5 + 26 =2 +3,332 1 =1 32 +3439.Rewriting a nested radical in this way is called denesting. This process is generally considered a dicult problem,although a special class of nested radical can be denested by assuming it denests into a sum of two surds:a bc =d e.Squaring both sides of this equation yields:a bc = d + e 2de.34 CHAPTER 2. NESTED RADICALThis can be solved by nding two numbers such that their sum is equal to a and their product is b2c/4, or by equatingcoecients of like termssetting rational and irrational parts on both sides of the equation equal to each other. Thesolutions for e and d can be obtained by rst equating the rational parts:a = d + e,which givesd = a e,e = a d.For the irrational parts note thatbc = 2de,and squaring both sides yieldsb2c = 4de.By plugging in a d for e one obtainsb2c = 4(a d)d = 4ad 4d2.Rearranging terms will give a quadratic equation which can be solved for d using the quadratic formula:4d24ad + b2c = 0,d =a a2b2c2.Since a = d+e, the solution e is the algebraic conjugate of d. If we setd =a +a2b2c2,thene =a a2b2c2.However, this approach works for nested radicals of the forma bc if and only if a2b2c is a rationalnumber, in which case the nested radical can be denested into a sum of surds.In some cases, higher-power radicals may be needed to denest the nested radical.2.1.1 Some identities of RamanujanSrinivasa Ramanujan demonstrated a number of curious identities involving denesting of radicals. Among them arethe following:[1]2.2. IN TRIGONOMETRY 543 + 2453 245=45 + 145 1=12(3 +45 +5 +4125),328 327 =13(398 328 1),353255275=5125+5325 5925,332 1 =319 329+349. [2]Other odd-looking radicals inspired by Ramanujan include:449 + 206 +449 206 = 23,3(2 +3)(5 6)+ 3(23 + 32)=10 13 565 +6.2.1.2 Landaus algorithmIn 1989 Susan Landau introduced the rst algorithm for deciding which nested radicals can be denested.[3] Earlieralgorithms worked in some cases but not others.2.2 In trigonometryMain article: Exact trigonometric constantsIn trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals. For example,sin60= sin 3=116[2(1 3)5 +5 +2(5 1)(3 + 1)]andsin24= sin 7.5=122 2 +3 =122 1+32.2.3 In the solution of the cubic equationNested radicals appear in the algebraic solution of the cubic equation. Any cubic equation can be written in simpliedform without a quadratic term, asx3+ px + q= 0,whose general solution for one of the roots is6 CHAPTER 2. NESTED RADICALx =3q2+q24+p327+3q2 q24+p327;here the rst cube root is dened to be any specic cube root of the radicand, and the second cube root is dened tobe the complex conjugate of the rst one. The nested radicals in this solution cannot in general be simplied unlessthe cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, wehave the casus irreducibilis, in which all three real solutions are written in terms of cube roots of complex numbers.On the other hand, consider the equationx37x + 6 = 0,which has the rational solutions 1, 2, and 3. The general solution formula given above gives the solutionsx =33 +103i9+33 103i9.For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet itis reducible (even though not obviously so) to one of the solutions 1, 2, or 3.2.4 Innitely nested radicals2.4.1 Square rootsUnder certain conditions innitely nested square roots such asx =2 +2 +2 +2 + represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign,which gives the equationx =2 + x.If we solve this equation, we nd that x = 2 (the second solution x = 1 doesn't apply, under the convention that thepositive square root is meant). This approach can also be used to show that generally, if n > 0, then:n +n +n +n + =12(1 +1 + 4n)and is the real root of the equation x2 x n = 0. For n = 1, this root is the golden ratio , approximately equal to1.618. The same procedure also works to get thatn n n n =12(1 +1 + 4n).and is the real root of the equation x2+ x n = 0. For n = 1, this root is the reciprocal of the golden ratio , whichis equal to 1. This method will give a rational x value for all values of n such that2.4. INFINITELY NESTED RADICALS 7n = x2+ x.Ramanujan posed this problem to the 'Journal of Indian Mathematical Society':? =1 + 21 + 31 + .This can be solved by noting a more general formulation:? =ax + (n + a)2+ xa(x + n) + (n + a)2+ (x + n) Setting this to F(x) and squaring both sides gives us:F(x)2= ax + (n + a)2+ xa(x + n) + (n + a)2+ (x + n) Which can be simplied to:F(x)2= ax + (n + a)2+ xF(x + n)It can then be shown that:F(x) = x + n + aSo, setting a =0, n = 1, and x = 2:3 =1 + 21 + 31 + .Ramanujan stated this radical in his lost notebook

5 +

5 +

5 5 +5 +5 +5 =2 +5 +15 652The repeating pattern of the signs is (+, +, , +)In Vites expression for piVites formula for pi, the ratio of a circles circumference to its diameter, is2=222 +222 +2 +22 .8 CHAPTER 2. NESTED RADICAL2.4.2 Cube rootsIn certain cases, innitely nested cube roots such asx =36 +36 +36 +36 + can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are leftwith the equationx =36 + x.If we solve this equation, we nd that x = 2. More generally, we nd that3n +3n +3n +3n + is the real root of the equation x3 x n = 0 for all n > 0. For n = 1, this root is the plastic number , approximatelyequal to 1.3247.The same procedure also works to get3n 3n 3n 3n as the real root of the equation x3+ x n = 0 for all n and x where n > 0 and |x| 1.2.5 See alsoSum of radicalsSpiral of Theodorus2.6 References[1] Landau, Susan. A note on 'Zippel Denesting'". CiteSeerX: 10 .1 .1 .35 .5512.[2] Landau, Susan. RADICALS AND UNITS IN RAMANUJANS WORK (POSTSCRIPT).[3] Landau, Susan (1992). Simplication of Nested Radicals. Journal of Computation (SIAM) 21: 85110. doi:10.1109/SFCS.1989.63496.CiteSeerX: 10 .1 .1 .34 .2003.2.6.1 Further readingLandau, Susan (1994). How to Tangle with a Nested Radical. Mathematical Intelligencer16: 4955.doi:10.1007/bf03024284.Decreasing the Nesting Depth of Expressions Involving Square RootsSimplifying Square Roots of Square RootsWeisstein, Eric W., Square Root, MathWorld.Weisstein, Eric W., Nested Radical, MathWorld.Chapter 3nth rootRoots of integer numbers from 0 to 10. Line labels = x. x-axis = n. y-axis = nth root of x.In mathematics, the nth root of a number x, where n is a positive integer, is a number r which, when raised to thepower n yields xrn= x,where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Rootsof higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc.For example:910 CHAPTER 3. NTH ROOT2 is a square root of 4, since 22= 4.2 is also a square root of 4, since (2)2= 4.A real number or complex number has n roots of degree n. While the roots of 0 are not distinct (all equaling 0), then nth roots of any other real or complex number are all distinct. If n is even and x is real and positive, one of its nthroots is positive, one is negative, and the rest are complex but not real; if n is even and x is real and negative, none ofthe nth roots is real. If n is odd and x is real, one nth root is real and has the same sign as x , while the other roots arenot real. Finally, if x is not real, then none of its nth roots is real.Roots are usually written using the radical symbol or radixor, withx orx denoting the square root,3xdenoting the cube root,4x denoting the fourth root, and so on. In the expressionnx , n is called the index,isthe radical sign or radix, and x is called the radicand. Since the radical symbol denotes a function, when a numberis presented under the radical symbol it must return only one result, so a non-negative real root, called the principalnth root, is preferred rather than others; if the only real root is negative, as for the cube root of 8, again the real rootis considered the principal root. An unresolved root, especially one using the radical symbol, is often referred to as asurd[1] or a radical.[2] Any expression containing a radical, whether it is a square root, a cube root, or a higher root,is called a radical expression, and if it contains no transcendental functions or transcendental numbers it is called analgebraic expression.In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:nx=x1/nRoots are particularly important in the theory of innite series; the root test determines the radius of convergence ofa power series. Nth roots can also be dened for complex numbers, and the complex roots of 1 (the roots of unity)play an important role in higher mathematics. Galois theory can be used to determine which algebraic numbers canbe expressed using roots, and to prove the Abel-Runi theorem, which states that a general polynomial equation ofdegree ve or higher cannot be solved using roots alone; this result is also known as the insolubility of the quintic.3.1 Etymology3.1.1 Origin of the root symbolThe origin of the root symbol is largely speculative. Some sources imply that the symbol was rst used by Arabicmathematicians. One of those mathematicians was Ab al-Hasan ibn Al al-Qalasd (14211486). Legend has itthat it was taken from the Arabic letter " " (m, /dim/), which is the rst letter in the Arabic word " " (jadhir,meaning root"; /dir/).[3] However, many scholars, including Leonhard Euler,[4] believe it originates from theletter r, the rst letter of the Latin word "radix" (meaning root), referring to the same mathematical operation.The symbol was rst seen in print without the vinculum (the horizontal bar over the numbers inside the radicalsymbol) in the year 1525 in Die Coss by Christo Rudol, a German mathematician.The Unicode and HTML character codes for the radical symbols are:3.1.2 Etymology of surdThe term surd traces back to al-Khwrizm (c. 825), who referred to rational and irrational numbers as audible andinaudible, respectively. This later led to the Arabic word " " (asamm, meaning deaf or dumb) for irrationalnumber being translated into Latin as surdus (meaning deaf or mute). Gerard of Cremona (c. 1150), Fibonacci(1202), and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots.[5]3.2 HistoryMain articles: Square root History and Cube root History3.3. DEFINITION AND NOTATION 113.3 Denition and notation+i i 1+1 The four 4th roots of 1,none of which is realAn nth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth poweris x:rn= x.Every positive real number x has a single positive nth root, called the principal nth root, which is writtennx . Forn equal to 2 this is called the principal square root and the n is omitted. The nth root can also be represented usingexponentiation as x1/n.For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nthroot. For odd values of n, every negative number x has a real negative nth root. For example, 2 has a real 5th root,52= 1.148698354 . . . but 2 does not have any real 6th roots.Every non-zero number x, real or complex, has n dierent complex number nth roots including any positive or negativeroots. They are all distinct except in the case of x = 0, all of whose nth roots equal 0.The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two12 CHAPTER 3. NTH ROOT0 +i i 1+1 The three 3rd roots of 1,one of which is a negative realnth powers) are irrational. For example,2 = 1.414213562 . . .All nth roots of integers, and in fact of all algebraic numbers, are algebraic.3.3.1 Square rootsMain article: Square rootA square root of a number x is a number r which, when squared, becomes x:r2= x.Every positive real number has two square roots, one positive and one negative. For example, the two square roots of3.3. DEFINITION AND NOTATION 130 2 4 6 8 10 123123The graph y= x .25 are 5 and 5. The positive square root is also known as the principal square root, and is denoted with a radicalsign:25 = 5.Since the square of every real number is a positive real number, negative numbers do not have real square roots.However, every negative number has two imaginary square roots. For example, the square roots of 25 are 5i and5i, where i represents a square root of 1.14 CHAPTER 3. NTH ROOT0 2 4 6 2 4 61212The graph y=3x .3.3.2 Cube rootsMain article: Cube rootA cube root of a number x is a number r whose cube is x:r3= x.Every real number x has exactly one real cube root, written3x . For example,38=2 and38= 2.Every real number has two additional complex cube roots.3.4 Identities and propertiesEvery positive real number has a positive nth root and the rules for operations with such surds are straightforward:nab =nanb ,3.5. SIMPLIFIED FORM OF A RADICAL EXPRESSION 15nab=nanb.Using the exponent form as in x1/nnormally makes it easier to cancel out powers and roots.nam= (am)1n= amn.Problems can occur when taking the nth roots of negative or complex numbers. For instance:1 1 = 1whereas1 1 = 1when taking the principal value of the roots.3.5 Simplied form of a radical expressionA non-nested radical expression is said to be in simplied form if[6]1. There is no factor of the radicand that can be written as a power greater than or equal to the index.2. There are no fractions under the radical sign.3. There are no radicals in the denominator.For example, to write the radical expression325in simplied form, we can proceed as follows. First, look for aperfect square under the square root sign and remove it:325=1625= 425Next, there is a fraction under the radical sign, which we change as follows:425=425Finally, we remove the radical from the denominator as follows:425=42555=4105=4510When there is a denominator involving surds it is always possible to nd a factor to multiply both numerator anddenominator by to simplify the expression.[7][8] For instance using the factorization of the sum of two cubes:13a +3b=3a23ab +3b2(3a +3b)(3a23ab +3b2)=3a23ab +3b2a + b.Simplifying radical expressions involving nested radicals can be quite dicult. It is not immediately obvious forinstance that:3 + 22 = 1 +216 CHAPTER 3. NTH ROOT3.6 Innite seriesThe radical or root may be represented by the innite series:(1 + x)s/t=n=0n1k=0(s kt)n!tnxnwith |x| < 1 . This expression can be derived from the binomial series.3.7 Computing principal rootsThe nth root of an integer is not always an integer, and if it is not an integer then it is not a rational number. Forinstance, the fth root of 34 is534 = 2.024397458 . . . ,where the dots signify that the decimal expression does not end after any nite number of digits. Since in this examplethe digits after the decimal never enter a repeating pattern, the number is irrational.3.7.1 nth root algorithmThe nth root of a number A can be computed by the nth root algorithm, a special case of Newtons method. Startwith an initial guess x0 and then iterate using the recurrence relationxk+1=1n((n 1)xk +Axn1k)until the desired precision is reached.Depending on the application, it may be enough to use only the rst Newton approximant:nxn+ y x +ynxn1.For example, to nd the fth root of 34, note that 25= 32 and thus take x = 2, n = 5 and y = 2 in the above formula.This yields534 =532 + 2 2 +25 16= 2.025.The error in the approximation is only about 0.03%.Newtons method can be modied to produce a generalized continued fraction for the nth root which can be modiedin various ways as described in that article. For example:nz=nxn+ y= x +ynxn1+(n 1)y2x +(n + 1)y3nxn1+(2n 1)y2x +(2n + 1)y5nxn1+(3n 1)y2x + ...;3.7. COMPUTING PRINCIPAL ROOTS 17nz= x +2x yn(2z y) y (12n21)y23n(2z y) (22n21)y25n(2z y) (32n21)y27n(2z y) ....In the case of the fth root of 34 above (after dividing out selected common factors):534 = 2 +140 +44 +6120 +94 +11200 +144 + ...= 2 +4 1165 1 4 6495 9 11825 14 161155 ....3.7.2 Digit-by-digit calculation of principal roots of decimal (base 10) numbersPascals Triangle showing P(4, 1) = 4 .Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there, x(20p +x) c, or x2+ 20xp c , follows a pattern involving Pascals triangle. For the nth root of a number P(n, i) is denedas the value of elementi in rown of Pascals Triangle such thatP(4, 1) =4 , we can rewrite the expression asn1i=010iP(n, i)pixni. For convenience, call the result of this expression y . Using this more general expression,any positive principal root can be computed, digit-by-digit, as follows.Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, asin long division, the root will be written on the line above. Now separate the digits into groups of digits equating tothe root being taken, starting from the decimal point and going both left and right. The decimal point of the root willbe above the decimal point of the square. One digit of the root will appear above each group of digits of the originalnumber.Beginning with the left-most group of digits, do the following procedure for each group:1. Starting on the left, bring down the most signicant (leftmost) group of digits not yet used (if all the digitshave been used, write 0 the number of times required to make a group) and write them to the right of theremainder from the previous step (on the rst step, there will be no remainder). In other words, multiply theremainder by 10nand add the digits from the next group. This will be the current value c.18 CHAPTER 3. NTH ROOT2. Find p and x, as follows:Let p be the part of the root found so far, ignoring any decimal point. (For the rst step, p = 0 ).Determine the greatest digit x such that y c .Place the digit x as the next digit of the root, i.e., above the group of digits you just brought down. Thusthe next p will be the old p times 10 plus x.3. Subtract y from c to form a new remainder.4. If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Oth-erwise go back to step 1 for another iteration.ExamplesFind the square root of 152.2756.1 2. 3 4 / \/ 01 52.27 56 01 10010012+ 10120111 1 < 10010022+ 10120121x = 1 01 y = 10010012+10120112= 1 + 0 = 1 00 52 10011022+ 1012112152 < 10011032+ 10121131x = 2 00 44 y = 10011022+ 10121121= 4 + 40 = 44 08 27 100112032+ 101212131 827 < 100112042+ 101212141x = 3 07 29y = 100112032+ 101212131= 9 + 720 = 729 98 56 1001123042+ 1012123141 9856 < 1001123052+ 1012123151x = 4 98 56 y = 1001123042+ 1012123141= 16 + 9840 = 9856 00 00 Algorithm terminates:Answer is 12.34Find the cube root of 4192 to the nearest hundredth.1 6. 1 2 4 3 / \/ 004 192.000 000 000 004 10010013+ 10130112+ 10230211 4 < 10010023+ 10130122+10230221x = 1 001 y = 10010013+ 10130112+ 10230211= 1 + 0 + 0 = 1 003 192 10011063+ 10131162+ 10231261 3192 < 10011073+ 10131172+ 10231271x = 6 003 096 y = 10011063+ 10131162+ 10231261= 216 + 1,080 + 1,800 = 3,096 096 000 100116013+ 101316112+ 102316211 96000