2d. What is the biquadrate root of 34827998976? Ans. 431·94. 3d. Extract the sursolid, or fifth root of 281950621875? Ans. 195. 4th. Extract the square cubed, or sixth root of 1178420166015625? Ans. 325. A GENERAL RULE FOR EXTRACTING ROOTS BY APPROXIMATION. 1. Subtract one from the exponent of the root required, and multiply half of the remainder by that exponent, and this product by that power of the assumed root, whose exponent is two less than that of the root required. The general theorem for the extraction of all roots, by approximation, from whence the rule was taken, and the Theorems deducible from it, as high as the twelfth power. Let G resolvend whose root is to be extracted. ‡r+e=root required; r being assumed as near the true root, and m-exponent of the power-then the equation will stand thus. By this Theorem the fraction is obtained in numbers to the lowest terms in all the odd powers; and in the even powers only by having the numerator and denominator found by this equation. 2. Divide the given number by the last product; and from the quotient subtract a fraction, whose numerator is obtained by subtracting two from the exponent, and multiplying the remainder by the square of the assumed root; and whose denominator is found by subtracting one from the exponent and multiplying the square of the remainder by the exponent. 3. After this subtraction is made, extract the square root of the remainder. 4. From the exponent subtract two, and place the remainder as a numerator; then subtract one from the exponent, and place the remainder under the numerator for a denominator. 5. Multiply this fraction by the assumed root; add the product to the square root, before found, and the sum will be the root required, or an approximation to it. EXAMPLE. What is the square cubed root of 1178420166015625 ? 6-1 Exponent of the required root is 6. Then, x6=15. 2 3001=8100000000 and this multiplied by 15=121500000000. 1178420166015625-121500000000=9698-9314, from this 325 43 And the sum is the approximated root= For the 2d operation, let 325-43 = assumed root. ANOTHER METHOD BY APPROXIMATION.* RULE. 1. Having assumed the root in the usual way, involve it to that power denoted by the, exponent less 1. A rational formula for extracting the root of any pure power by approximation. Let the resolvend be called G, and let r+e be the required root, r being as sumed in the usual way. C c --. 2. Multiply this power by the exponent. 3. Divide the resolvend by this product, and reserve the quotient. 4. Divide the exponent of the given power, less 1, by the exponent, and multiply the assumed root by the quotient. 5. Add this product to the reserved quotient, and the sum will be the true root, or an approximation. 6. For every succeeding operation, let the root last found, be the assumed root. EXAMPLE. What is the square cubed root of 1178420166015625 ? The exponent is 6. Let the assumed root be 300. Then 3005×6=14580000000000 14580000000000) 1173420166015625(80-824. Add x 300 250 330-824 approximated root. For the next operation, let 330 824 be the assumed root. SURDS. I. SURDS are quantities, whose roots cannot be obtained exactly, but may be approximated to any definite extent by continuing the extraction of the roots. Surds are expressed by fractional indices or exponents, or by the radical sign. Thus, 3a, or √3, denotes the square root of 3. The value of 2 or 2, to the hundreth place of decimals, is 141, and to the millionth place is 1.414213. The value of 2 may be obtained more nearly by continuing the extraction, but can never be obtained with perfect accuracy, as is easily proved in the following section. Surds are often called irrational quantities, because their value cannot be expressed by figures. They are thus distinguished from assignable quantities, which are called rational quantities. Thus, 2 is a rational, and √2, an irrational quantity. A surd is always connected with a rational quantity expressed or understood. Thus, as the square root of 2, or 2, is that quantity taken once, unity is understood, and the surd is expressed either √2, 1√2, or 1X√2. If the surd is to be taken more than once, the number of times is always expressed; thus 3√3, or 3×√3, means thrice the square root of 3, or the surd taken three times. 3 Hence it is that an expression of this form, 1/2, or 35, is considered as consisting of 2 parts, a rational, and an irrational part, the rational part always expressing the number of times the surd is taken. 5 4 3 3 From the notation of powers, and surds, these expressions are equivalent; viz. 3=35; and 2√22. Also, 553, that is, the square root of the cube of 5, or the cube of the square root of 5. Note. Though surds are expressed by means of fractional indices or the radical sign, yet it is common to apply the same indices or radical sign, to complete powers, whose roots are to be extracted. The student will observe, therefore, that quantities expressed in the form of surds are not necessarily surd quantities. One number also may be a complete power of one kind, but not of another. Thus 4 is 2, but 4 is a surd; and 64 is 8, and 643 is 4, but 644 and 64 or 64 and 64 are surds. 3 5 II. As few numbers are complete powers, surds must very often occur in arithmetical operations. If the root of a whole number is not a whole number, neither is it a whole number and a decimal, which can be assigned. For, supposing the entire root to be obtained, when it was raised to the power, it would produce a whole number and a decimal; while the supposition requires that only a whole number should be produced. Thus, supposing the square root of 2, or √2, to be exactly 141, or 1-414213, this root raised to the square should produce 2; but it is obvious that the square would be a whole number and a decimal, and not the number 2. It is equally evident, that the root of a vulgar fraction cannot be assigned, unless both parts of the fraction when reduced to its lowest terms, are complete powers of the roots required. Thus √√; but is a surd, and the entire value of the square root of the fraction cannot be obtained. III. Though the value of a surd cannot be assigned, its power is assignable. From the definition of a root, it is evident that 22 or 2 is such a number as multiplied by itself, the product or square will be 2. Thus √2x√2 or 2x2=2. And 3x3x3= 3, and thus for other surds. IV. Arithmetical calculations are often simplified by certain operations on surds, or quantities in the form of surds. Rules for several operations follow. 1. Any number may be reduced to the form of a surd, by raising it to the power denoted by the index of the surd, and then placing the power under the radical sign. Thus to reduce 2 to the form of the square root; because 2×2=22=4, 2=√/22=√4. Reduce 2 to the form of the fifth root. 3 Ans. $32. Ans. 125. Ans. Reduce 5 to the form of the third root. Reduce 7 to the form of the fourth root. 2. Surds are reduced to their most simple terms, by resolving the quantity under the radical sign into two factors, one of which shall be a complete power of the given root; and then placing the root of this power before the other factor under the radical sign. Thus √27=√9×3=√9×√3=3×√3 or 3√3. Also,√32=√16x2=16 ×√2=2√2. 3 4 Reduce 481, √351, and √896 to their most simple terms. Ans. 10√5. Hence, it is obvious, that if a factor be multiplied into a surd, the whole may be reduced to the form of a surd, by raising the factor to the power denoted by the surd, multiplying the power into the surd, and placing the product under the radical sign. Thus 3√3=√32X√3=√9X3=√27; and 8V7=V83XV7=√512X7= 3 √3584. 3. Surds of the same radical sign may be added together, when the quantities under the radical sign are the same, by prefixing the sum of the rational parts to the surd quantity. Thus 1/2+1/2, or √2+√2=2√2, or twice √2; and 3√5+45=7V5. 3 If the surds are not already in their most simple terms, they may often be added after the reduction is made. Thus √20+√80= 2√5+4√5=6√5; and, 162+1350=32+5√2=8√2. 3 What is the sun of √56 and √3584 ? Ans. 10/7. 4. Surds of the same radical sign may be subtracted, if the surd part be the same, by placing the difference of the rational parts before the surd. If the quantities are not already in their simplest |