Roots of Numbers expressed in Decimals. 6. Find the square root of 0,000256. 7. Find the square root of 2. 8. Find the square root of 13. Ans. 0,016. Ans. 11. 146. Corollary. When there is any remainder after the processes of arts. 142 and 145, it shows that the given numbers are not exact powers; so that the roots obtained, instead of being the roots of the numbers themselves, are those of the greatest squares contained in the numbers. But by the annexing of cyphers to the right of the given number, decimal places may be obtained in the root, and the roots thus found will differ less and less from the required root. EXAMPLES. 1. Find the 3d root of 1345 to two places of decimals. Solution. The operation is as follows: 1.345,000.000 11,03+ Ans. 1 3/3 1331 1|3,6 1331,000 14|3,6 1341,919727 3,080273 Remainder. 1 Roots of Numbers expressed in Decimals. 2. Find the square root of 5 to 3 places of decimals. 3. Find the square root of 101 to 3 places of decimals. Ans. 10,049 +. 4. Find the square root of 9,6 to 3 places of decimals. Ans. 3,098 +. 5. Find the square root of 0,003 to 5 places of decimals. Ans. 0,05477+. 6. Find the 3d root of 12 to 3 places of decimals. Ans. 2,289 +. 7. Find the 3d root of 28,25 to 3 places of decimals. 147. Corollary. The roots of vulgar fractions and mixed numbers may in the same way be computed in decimals by first reducing them to decimals. EXAMPLES. 1. Find the square root of to 4 places of decimals. Ans. 0,2425+. 2. Find the square root of to 3 places of decimals. Ans. 0,645 +. 3. Find the square root of 1 to 2 places of decimals. Ans. 1,32+. 4. Find the square root of 11 to 3 places of decimals. Ans. 3,418+. 5. Find the 3d root of to 3 places of decimals. Ans. 0,873 +. Ans. 0,941 + 6. Find the 3d root of to 3 places of decimals. A Fraction is not the Root of an Integer. 7. Find the 3d root of 15% to 3 places of decimals. Ans. 2,502 +. 148. Scholium. It might be thought that, though a given integer has no exact integral root, it still may have an exact fractional root, which is not obtained by the preceding pro cess. But this is readily shown to be impossible, for suppose the fractional root, when reduced to its lowest terms, to be A B' the nth power of this root is An since A and B have no common divisor, and since every prime number which divides A" must divide A, and every prime number which divides B" must divide B, it follows that there is no prime number which divides both A and B, and, therefore, A and B have no common divisor; so that the fraction An is already reduced to its lowest terms, and cannot be an integer. SECTION V. Binomial Equations. 149. Definition. When an equation with one unknown quantity is reduced to a series of monomials, Solution of Binomial Equations. and all its terms which contain the unknown quantity are multiplied by the same power of the unknown quantity, it may be represented by the general form AxM0, and may be called a binomial equation. 150. Problem. To solve a binomial equation. Solution. Suppose the given equation to be Hence, find the value of the power of the unknown quantity which is contained in the given equation, precisely as if this power were itself the unknown quantity; and the given equations are of the first degree. Extract that root of the result which is denoted by the index of the power. 151. Corollary. Equations containing two or more unknown quantities will often, by elimination, conduct to binomial equations. EXAMPLES. 1. Solve the two equations xy7+2y7-4 y3-8x+16 = 0, x2y74 y7 4 x y3 + 8 y3 +32 x — 64 = 0. - Examples of Binomial Equations. Solution. The elimination of y between these two equations, by the process of art. 116, gives being substituted in the first of the given equations, produces shown when we treat of the theory of equations. Again, the value of x, x= being substituted in the first of the given equations, produces whence we have જીજે 8, 4 y3 +32=0, y = 2 or — 1 ± √3, as will be shown in the theory of equations. 2. Solve the equation 3 x2 + 2 x = x2 + 2 x + 18. Ans. x = ± 3. |