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? 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, 3, or 3, denotes the square root of 3. The value of 24 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 1×2. If the surd is to be taken more than once, the number of times is always expressed; thus 3/3, or 3X√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 3√5, 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. 3 From the notation of powers, and surds, these expressions are equivalent; viz. 3√35; and 2√2. Also, 5/53, 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 6 1 and 643 4 5 or √64 and 64 are surds. 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 21×2=2. And 33×3×33= 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 2x2=22=4, 2=√√/22=√/4. Reduce 2 to the form of the fifth root. 5 Ans. √32. 3 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=√9X3=√9X√3=3×√3 or 3√3. Also,√32=√16x2=16 XV2=22. 4 Ans. 55. 3 Ans. 8√7. Ans.. 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=√33×√3=√9X3=√27; and 3√7=√83XV7=√512x7= √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 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=3√2+5V2=8√2. 3 3 What is the sum of 56 and √3584? 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 terms, they should be reduced to this form. Thus 4-4=0; and 3√3—2√3=1√3 or √3. Also 75--45-35. What is the difference between 5 1350 and 162? Ans. 22. Note 1. Surds, apparently incapable of addition or subtraction except by their signs, may sometimes be reduced to a common surd, by the following process, and their sum and difference readj. ly found. Thus let the surds be v and V. As √√, and as √√√, then + } = 2 √ } + { √ } = W==√6=√6, their sum: And 2 6, their difference. What is the sum and difference of √ and √? Ans. Their sum is 6. Their diff. is√6. 12 Ans. ✓. Ans. ¿√15. What is the sum of 15 and √? What is the difference of ✔ and ? Note 2. If the same quantity is under different radical signs, or if the same radical sign bas different quantities under it when the surds are in their simplest terms, the surds can be added or subtracted only by the signs of addition or subtraction. Thus it is evident, that √2+2, is neither twice the square root of 2 nor twice the cube root of 2; and that 3√3—2√3, is neither the square root of 3 nor the cube root of 3. It is equally obvions, that 2√3 +2√2, is neither four times the square of 3 nor of 2; and that 4√2 -2√3, is neither twice the square of 2 nor of 3. 3 3 3 5. Surds of the same radical sign are multiplied like other numbers, but the product must be placed under the same radical sign. Thus √27×64-27×64=√1728=12, for 27=3, and 64 =4, and √27x64=3x4=12. And √2×√3=√2×3=√6. Also 3√3X4√5=12/15 or √27X√80, and √27X√80=√27X80 =12/15=√2160. Sometimes the product of the surds becomes a complete power of that root, and the root should then be extracted, as in the first of the preceding examples. Also in this example; √2X√200= ✓400=20. It is evident from the first example in this section, that, when quantities are under the same radical sign, the root of the product of quantities is equal to the product of their roots. If a surd be raised to a power denoted by the index of the root, the power will be rational. Thus, √3X√3, or 31×3=3. In this example 2 is the index of the root, and the surd is raised to the second power or square. Also V4x4x44. If fractional indices be used, in order to multiply surds of the same root, you 3 3 3 1 1 3 have only to add the indices. Thus 43×45×13=48=41 or 4, unity being the implied index of 4, or of the first power of any numIn all cases when the sum of the numerators contains the com ber. mon denominator a certain number of times exactly, the product will 2+2+2 2 6 be rational. Thus 3x33x33 ===3} = 5+5 10 5 5 3 =33=32=9, and 72x72 As 54 may be expressed according to the notation of powers, 3 4+3 51×5T=5 3 thus, 51, and 53 by 5, hence 5x5-551-57. Therefore, to multiply different powers of the same root, you have only to add the indices of the given root, and place the sum for the index of the power which is produced. Thus 32x32=34, or the square of a number multiplied by itself produces the fourth power; the cube by the cube, the sixth power, and so on. Thus also 2×2a= 8 2i+1=24=2i=22. 3 5 3 = 6. Surds of the same radical sign are divided like whole num. bers, but the quotient must be placed under the same radical sign. Thus 1728-64=√1728÷64=√27 = 3; and √/6÷÷√√/3= √6-3=√2. Sometimes the quotient becomes a complete power, as in the first example, in which case the root should be extracted. So also in the following; √400÷÷√100=√400÷÷÷100=√4=2. As 1728=12, and 641, then 1728÷÷√64=12÷4=3= √27. Hence, the quotient of the roots of quantities is the same as the root of their quotient, if the quantities are under the same radical sign. Divide 103 by 6. Now √108÷√C=√ 108÷÷6= √18= √9×2=3√2. Divide 100 by 32. Now 9/100=8100, and 3/2= 18, and 8100√18=√810018=√450=15√2. Or 9√100 3/28÷3 × √ 100÷÷÷√2 = 9÷÷3 × √ 100÷29÷÷3 × √ 50≈ 3√50=3x5√2=15√2. 3 Divide by 1. Now }÷}=}; and √ √÷V != ↓ M Divide 48 by √-; 5√60 by S√15; and 4√ by !√}. If the quantities under the radical sign be the same, the quotient will be found by dividing the rational parts only. Thus √2÷√2 ==√11, or 2 is contained in 2 once. Also √3÷√3=2, and 2√✓5+55=;• To divide one power by another of the same root, place the difference of the indices for the index of the given root. This is merely reversing a process given in the preceding section. The reason of the process may also be seen in the following manner. Thus 29×24 20 |