The surd 7/ a .x q a 7/ ; and, in Eke manner, if a power of any quantity of the same name with the surd divides the quantity under the radical sign without a remainder, as here am divides am x, and 25, the square of 5, divides 75, the quan tity under the sign in t/73, without a re mainder; then place the root of that pow er rationally before the sign, and the quo tient under the sign, and thus the surd will be reduced to a more simple expres sion. Thus, a? 75 7-= 5 %/ V 48 = .1/3X 16 = 4 ; 81 = 13/27 x 3 = 3 .4/ 3.
When surds are reduced to their least expressions, if they have the same irra tional part, they are added or subtracted, by adding or subtracting their rational co efficients, and prefixing the sum or dif ference to the common irrational part. Thus, V75 + V48 5 + = 9 ,/ 3; 81 + 24= 3 + 2 4/3 = 54/3; s/150-4/54=5,/ 6-3 2 ,/ 6; VT. s= a %,/ x + b %/ x=a + %/ x.
Compound surds are such as consist of two or more joined together; the simple surds are commensurable in power, and by being multiplied into themselves, give at length rational quantities; yet com pound surds, multiplied into themselves, commonly give still irrational products; But when any compound surd is pro posed, there is another compound surd, which, multiplied into it, gives a rational product. Thus, if ,/ a + %/ b were pro posed, multiplying it by ./ a — b, the product will be a — b.
The investigation of that surd, which, multiplied into the proposed surd, gives a rational product, is made easy by three theorems, delivered by Mr. Maclaurin, in
his Algebra.
This operation is of use in reducing surd expressions to more simple forms. Thus, suppose a binomial surd divided by another, as V 20+ V 12, by V 5 — V 3, the quotient might be expressed by ./ 20 + 12 But this might be ex pressed in a more simple form, by multi plying both numerator and denominator by that surd, which, multiplied into the de nominator, gives a rational product : thus, V 5 5 ? 3+ = 10U 15 — + 8-' 5-3 2 2./ 15.
When the square root of a surd is re quired, it may be found, nearly, by ex tracting the root of a rational quantity that approximates to its value. Thus, to find the square root of 3 + 2 ,/ 2, first calcu late 2 = 1,41421. Hence 3+2 2 = 5,82842, the root of which is found to be nearly 2,41421.
In like manner we may proceed with any other proposed root. And if the in dex of the root proposed to be extract ed be great, a table of logarithms may be used. Thus, ,:/5 + 17 may be most conveniently found by logarithms.
Take the logarithm of 17, divide it by 13 ; find the number corresponding to the quotient ; add this number to 5: find the logarithm of the sum, and divide it by 7, and the number corresponding to this quo tient will be nearly equal to But it is sometimes requisite to express the roots of surds exactly by other surds. Thus, in the first example, the square root of 3+2 %/ 2 is 1 + %/ 2: for 1+,/ 2 X 1+,/ 2 = 1 + 2+2 = 3 +2 ./ 2. For the method of performing this, the curious may consult Mr. Maclau rin's Algebra.