2a36aab+3abb) 6a+bb—20a3b3 +15aab1 6a4bb 18a3b39aab 2a3. 6aab+6abb-b3) — 2α3b3 +6αab1 2a3b3+6aab 6ab5+b 0. 283. We easily deduce from the rule which we have explained, the method which is taught in books of arithmetic for the extraction Some examples in numbers: of the square root. 284. But when there is a remainder after the whole operation, it is a proof that the number proposed is not a square, and consequently that its root cannot be assigned. In such cases, the radical sign, which we before employed, is made use of. It is written before the quantity, and the quantity itself is placed between parentheses, or under a line. Thus, the square root of a a+bb is represented by (a a+bb), or by vaa+bb; and (1 − x x), or √1—xx, expresses the square root of 1 xx. Instead of this radical sign, we may use the fractional exponent, and represent the square root of a a+bb, for instance, by (a a+bb), or by a a+bb) }. CHAPTER VIII. Of the Calculation of Irrational Quantities. 285. WHEN it is required to add together two or more irrational quantities, this is done, according to the method before laid down, by writing all the terms in succession, each with its proper sign. And with regard to abbreviation, we must remark that instead of ✔a+va, for example, we write 2√ã; and that ✔ã vā = 0, because these two terms destroy one another. Thus, the quantities 3+√2 and 1 +√2, added together, make 4 + 2 √2 or 4 + √8; the sum of 5+√3 and 4 — √3 is 9; and that of 2√3 + 3√2 and √3 — √2 is 3√3 + 2√2. a 286. Subtraction also is very easy, since we have only to add the proposed numbers, changing first their signs; the following example will show this; let us subtract the lower number from the upper. 4- √2+2 √3 −3 √5 +4√6 - 3 — 3√2 +4 √3 + 2 √5 − 2 √б 287. In multiplication we must recollect that a multiplied by Va produces a; and that if the numbers which follow the sign✔ are different, as a and b, we have ab for the product of a multiplied by. After this it will be easy to perform the following examples: 288. What we have said applies also to imaginary quantities; we shall only observe further, that a multiplied by a produces a. If it were required to find the cube of 13, we should take the square of that number, and then multiply that square by the same number: see the operation: −1+3 1--3 --3-3 1-2-3-3--2-2-3 -1+3 2+2 3 -2√3+ 6 268. 289. In the division of surds, we have only to express the proposed quantities in the form of a fraction; this may be then changed into another expression having a rational denominator. For if the denominator be a + b, for example, and we multiply both it and ✔, the new denominator will be a a - b, the numerator by a in which there is no radical sign. Let it be proposed to divide 3+22 by 1+2; we shall first have 3+2√2 Multiplying √2, we shall have for the numerator: I Our new fraction therefore is ——1; and if we again mul tiply the terms by 1, we shall have for the numerator √2 + 1, and for the denominator +1. Now it is easy to show that √2 + 1 3+2√2 is equal to the proposed fraction 1+ √2 +1 being mul tiplied by the divisor 1 + √, thus, 8 1 + 2 √2 + 2 = 3 + 2 √ē. Another example: 852 divided by 3-2√2 makes 5/2 3-2√2 Multiplying the two terms of this fraction by 3 + 2√2, Consequently the quotient will be 4 +2. The truth of this may be proved in the following manner : 4+ √2 12 + 3√2 -8√2-4 12-52-48-5√2 290. In the same manner, we may transform such fractions into others, that have rational denominators. If we have, for example, 291. When the denominator contains several terms, we may in the same manner make the radical signs in it vanish one by one. Let 1 the fraction V10 √3 be proposed; we first multiply these terms by 10+√2 + √3, and obtain the fraction 5-246 Then multiplying its numerator and denominator by 5 + 26, we have 510 11√2 + 9 √3 +260. |