Note. It is important to arrange the dividend and divisor in the same order of powers, and to keep this order throughout the work. 15. m - 3m3n+3mn-n by mnt. -5 16. xy - 3x-y+xy by x-2y+x-3y-4 - x+y-5. 17. a + ab + b by a - ab+b2. -2 18. m+mn2+n by m1+mn-1+mn-2. 226. We will now prove that Rule II., Art. 216, holds for all values of m and n. We will consider three cases, in each of which m may have any value, positive or negative, integral or fractional. CASE I. Let n be a positive integer. Then, from the definition of a positive integral exponent, CASE II. Let n be a positive fraction, which we will de (am)= = √(am), by the definition of Art. 218, = Vamp, by Case I., CASE III. Let n be a negative quantity, which we will denote by - s. Then, 1 (am)-= by the definition of Art. 221, = (am)' 1 ams by Cases I. or II., =am(-s). We have therefore for all values of m and n, 1. (α2)-8. 227. Find the values of the following: 5. (x2)-2. 9. (m3). 13.() 2. (α-2)2. 6. (a-1). 10. (y-3)-5. 14. 1 n3 3. (a). 7. (a). 11.(). 2 15.[(x)2] 2n 4. (c). 8. (√x). 12. (2). 16. (am)m-n. 228. To prove that (ab)" = a"b", for any value of n. In Art. 193 we showed the truth of the theorem for a positive integral value of n. CASE I. Letn be a positive fraction, which we will denote CASE II. Let n be a negative quantity, which we will XX. RADICALS. 230. A Radical is a root of a quantity indicated by a radical sign; as, Va, or √x+1. If the indicated root can be exactly obtained, it is called a rational quantity; if it cannot be exactly obtained, it is called an irrational or surd quantity. 231. The degree of a radical is denoted by the index of the radical sign; thus, √x + 1 is of the third degree. 3 5 232. Similar Radicals are those of the same degree, and with the same quantity under the radical sign; as, 2Vax and 3 Vax. 5 233. Most problems in radicals depend for their solution on the following important principle (Art. 228): TO REDUCE A RADICAL TO ITS SIMPLEST FORM. 234. A radical is in its simplest form when the quantity under the radical sign is not a perfect power of the degree denoted by any factor of the index of the radical, and has no factor which is a perfect power of the same degree as the radical. CASE I. 235. When the quantity under the radical sign is a perfect power of the degree denoted by a factor of the index. 1. Reduce 8 to its simplest form. 8=2=2 = 2 = √2, Ans. |