SCHOLIUM: The application of the rules of this chapter to the resolving of equations, shall be explained in the succeeding chapters, which treat of the solution of the different classes of them ; but some examples of their use in preparing equations for a solution, are the following: If a member of an equation be a furd root, then the equation may be freed from any surd, by bringing that member first to stand alone upon one side of the equation; and then taking away the radical sign from it, and raising the other side to the power denominated by the index of that surd. This operation becomes a necessary step towards the solution of an equation, when any of the unknown quantities are under the radical sign. Example. If 31x—a? +2y=aty 3 a=amy And 9x*-=a?~2ay+y?. If the unknown quantity be found only under the radical sign, and only of the first dimension, the equation will become simple, and may be resolved by the preceding rules. lills 19 riviob'g'r Then 4*+16** 1'3 ton And 48+16=64 If the unknown quantity in a final equation has fractional exponents, by means of the preceding rules a new equation may be fubstituted, in which the exponents of the unknown quantity are integers. Thus, if x++3x}= *+3x}=10, by reducing the furds to the same denomination, it becomes x +387 =10; and if x=x+, then x'+3x4, =10; and if this equation be resolved, N from 4 from a value of %, a value of x may be got by the rules of the next chapter. Thus also, if x+ 2x -2x1—3x3 =100. If x==%, this equation becomes z' +27-32°=100. P m In general, if x? + **=a, by reducing the surds to the same denomination, *m? + pn mg * 9" =a, and if xqn=x, then the equation is " +% =a, in which the exponents of z are integers; and z being found, x is to be found from the equation 49"=%. 5 QUATIONS were divided into orders according to the highest index of the unknown quantity in any term. (Chap. III.). Equations are either pure or adfested. Def. 1. A pure equation is that in which only one power of the unknown quantity is found. 2. An adfe&ted equation is that in which different powers of the unknown quantity are found in the several terms. Thus, a? tax?=b, axa-b?=m2 + x2, are pure equations. And xamax=6, *? *-* = 17, are adfected. I. 1. Solution of Pure Equations. Rule. Make the power of the unknown quantity to stand alone, by the rules formerly given, and then extract the root, of the same denomination, out of both sides, which will give the value of the unknown quantity. The index of the power may also be fractional; as, in the last example, m may be any number whatever. Let m=, then, as before, |