325. In order to form a complete theory of Indices it would be necessary to give demonstrations of several propositions which will be found in the larger Algebra. But these propositions follow so naturally from the definitions and the properties of fractions, that the student will not find any difficulty in the simple cases which will come before him. We shall therefore refer for the complete theory to the larger Algebra, and only give here some examples as specimens. 326. If m and n are positive whole numbers we know that (am)=am; see Art. 279. Now this result will also hold when m and n are not positive whole numbers. For example, For let (a)=x; then by raising both sides to the fourth power we have a=a; then by raising both sides to the third power we have a = 12; therefore x=a which was to be shewn. 327. If n is a positive whole number we know that a" × b"=(ab)". This result will also hold when n is not a positive whole number. For example, axba= (ab)§. For if we raise each side to the third power, we obtain in each case ab; so that each side is the cube root of ab. Suppose now that there are m of these quantities a, b, c,..., and that all the rest are equal to a; thus we obtain 1 (a)" _ (am)"; that is, (~/a)" = "Ja". = Thus the mth power of the nth root of a is equal to t1 nth root of the mth power of a. 328. Since a fraction may take different forms without any change in its value, we may expect to be able to give different forms to a quantity with a fractional index, without altering the value of the quantity. Thus, for example, we may expect that a3 = a; and this is the 2 since 4 6 case. For if we raise each side to the sixth power, we obtain a1; that is, each side is the sixth root of a1. 329. We will now give some examples of Algebraical operations involving fractional and negative exponents. +1 Here in the first line x3×x=x+1=x3, x3×x3=x3, ×x ̄=xo=1; and so on. 1. 9 ̄§. 2. 4 ̄a. 3. (100) ̄§. 4. (1000)3. 5. (81) ̄†. 16. a1-2+a by a-a ~17. a+a3b3—x3y3 by a+a3b3+x3y3. 18. xa—xy1+x3y—y# by x+x1yś+y. 23. a3+a313+b3 by aa+a*b*+ba. 26. xa—4x3y3+6xay1—4x3y§+y3 by x‡− 2x*y* +y1. Find the square roots of the following expressions. 27. x2-4+4x-1. 28. (x+x-1)-4(x-x-1). 29. x3-4x3+2x3 + 4x − 4x3 + x3. 30. 4x-12x+25-24x+16.x. XXXIV. Surds. 330. When a root of a number cannot be exactly obtained it is called an irrational quantity, or a surd. Thus, for example, the following are surds; And if a root of an algebraical expression cannot be denoted without the use of a fractional index, it is also called an irrational quantity or a surd. Thus, for example, the following are surds; α Ja, √ √(a2+ab+b2), Ya2, Y(a3+b3). The rules for operations with surds follow from the propositions of the preceding Chapter; and the present Chapter consists almost entirely of the application of those propositions to arithmetical examples. 331. Numbers or expressions may occur in the form of surds, which are not really surds. Thus, for example, 9 is in the form of a surd, but it is not really a surd, for √93; and √(a2+2ab+b2) is in the form of a surd, but it is not really a surd, for (a2+2ab+b2)=a+b. 332. It is often convenient to put a rational quantity into the form of an assigned surd; to do this we raise the quantity to the power corresponding to the root indicated by the surd, and prefix the radical sign. For example, 3 = √√32 = √9; 4= 3/43= 3/64; a= a; a+b= (a+b). 333. The product of a rational quantity and a surd may be expressed as an entire surd, by reducing the rational quantity to the form of the surd, and then multiplying; see Art. 327. For example, 3√√2= √√√9 × √√√2= √18; 23/4 3/8 × 3/4 = √/32; a√b= √a2 × √b= √(a2b). 334. Conversely, an entire surd may be expressed as the product of a rational quantity and a surd, if the root of one factor can be extracted. For example, √32 = √(16 × 2) = √16 × √2=4√2; 3/48=3/(8 × 6) = 3/8 × 3/6=23/6; 31a3b2)=2/a3× 3/b2=a3/b2. |