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 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 Ax2 + M = 0, 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 x y2+2y7-4 y3-8x + 16 = 0, y3 x2y7-4 y 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, pro shown when we treat of the theory of equations. Again, the value of x, x= 2, being substituted in the first of the given equations, pro as will be shown in the theory of equations. 2. Solve the equation 3x2 + 2 x = x2 + 2 x + 18. Ans. x=3. 9. Solve the two equations y6-33 y3x4 — 17 x2 = 0, -- yo+17 y3x433 x2 = 0. Ans. x = 0, and y = 0; or x =±5, and y = 2. 10. What number is it, whose half multiplied by its third part, gives 864 ? Ans. 72. Examples of Binomial Equations. 11. What number is it, whose 7th and 8th parts multiplied together, and the product divided by 3, gives the quotient 2983? Ans. 224. 12. Find a number such, that if we first add to it 94, then subtract it from 94, and multiply the sum thus obtained by the difference, the product is 8512. Ans. 18. 13. Find a number such, that if we first add it to a, then subtract it from a, and multiply the sum by the difference, the product is b. (a2b). Ans. 14. Find a number such, that if we first add it to a, then subtract a from it, and multiply the sum by the difference, the product is b. Ans. (a2 + b). 15. What two numbers are they whose product is 750, and quotient 34 ? Ans. 50 and 15. 16. What two numbers are they whose product is a, and quotient b? a Ans. a b and b 17. What two numbers are they, the sum of whose squares is 13001, and the difference of whose squares is 1449? Ans. 85 and 76. 18. What two numbers are they, the sum of whose is a, and the difference of whose squares is b? squares Ans. (a+b) and (a - b).. 19. What two numbers are to one another as 3 to 4, the sum of whose squares is 324900? Ans. 342 and 456. 20. What two numbers are as m to n, the sum of whose squares is a? Ans. m↓ a and nNo a √(m2 + n2) √ (m2 + n2) 21. What two numbers are as m to n, the difference of |