Reduce a and b to surds with the same radical sign. ANS. /a and /b2. Required the product of a multiplied by a*. ANS. /a". Find the sum and difference of √25ab and 16a2b. CHAP. XXX. OF THE CALCULATION OF IRRATIONAL QUANTITIES. 246. WHEN it is required to add together two or more irrational quantities, it is done according to the method already shown, by writing in succession all the terms, with the proper signs prefixed to each; and it is only necessary to remark, with respect to abbreviations, that instead of Na+a we may write 2a, and that a-a will of course be equal to 0, since these two terms destroy each other. Thus the quantities 3 + √2 and 5 + √2 added together will be 8+2√2, or 8+8; that the sum of 5+ √3 and 4-√√3 will be 9; and that the sum of 23, 32, and √3 −√2 is 3√3+2√2, or √27+ √8. 247. Subtraction is equally easy, since we have only to follow the same rules as have been laid down for the subtraction of rational quantities, viz. to change the signs of the quantities to be subtracted, and to proceed as in addition; as in the following example. From Take Rem. 4- √2+2√3−3√5+4√6 3-32+4/3+25-2/6. 248. In Multiplication, we must recollect that a × √a=a; and that if the numbers to which the radical sign is prefixed be different, as a and b, the product of a × √b will be √ab. It will therefore be easy to calculate the following examples; 249. What we have observed is likewise equally applicable to impossible or imaginary quantities, since we have only to bear in mind that a × √ If therefore it were required to find the cube of -1+ √-3, by proceeding in the ordinary method, the process will be as follows, viz. 250. In the division of irrational quantities, we have only to express the quantities proposed in the form of a fraction, which fraction may be afterwards changed into another, with a rational denominator. For example, if the denominator be a + b, and it be multiplied by a-b, we shall have a2-b for a new denominator, in which there is no radical sign. Suppose, for example, that it were required to divide 3+2/2 by 1+ 2; we shall first have the fraction, √b, Multiplying each term by 1-2, we shall have for the And if each term be now multiplied by 1. we shall have √2+1 1 , or √2+1; and it is easy to show that this quantity is equivalent to the fraction 3+2√2 ; for if √2+1 be multiplied by 1 + √2, we shall obtain the following product, √2+ 1 1 + √2 √2+1 +2+√2 √2+3+√2=3+2√2. Again, if it were required to divide 8 - 5/2 by 3 −2√2, which, expressed fractionally, is 8-5√2 3-2√2 if we multiply each term of the fraction by 3+2√2, we shall obtain the following new fraction, and we prove immediately by trial that 4+√2x3-√2-8-5√2, and consequently that 4+2 is the exact quotient of 8-5/2 divided by 3-2√2. 251. In like manner may any fraction, whose denominator is irrational, be transformed into one with a rational denomi In the same manner, if the denominator contains several terms, we may make the radical signs vanish by degrees. Let the fraction 1 √10-√2-3' be proposed for example. Mul tiplying √10+ √2+ √3, we shall have first and multiplying again by 5 +2√6, we shall have 5√10+11√2 +9√3 +4√15 1 |