The root of the first factor 36 ab can be found exactly, and the expression becomes 6 a b2 (2 a b c)*. This expression is much more simple than the other, for now it is necessary to find the root of only 2 a b c. The expression might have been put in this form, ¿24 c‡ – 6.2a (72)* a*b* c* = (36.2) a14 2+ c = 6 a b2 (2 a b c). Examples. = 6.2 a a‡ ba b‡ c‡ 1. Reduce (16 a' b′)3 to its simplest form. Ans. 2 a b (2 aa b)*. 2. Reduce (54 a x') to its simplest form. 4. Reduce (16 a3 b' + 32 a b3 m) to its simplest form. (16 aa b3 + 32 aa b3 m)* = (16 a' b2)1 (ab3 + 2 b m)a a*x. α Ans. 4 a b (a b3 +2b m) a7 x c3) 5. Reduce (135 a'r -- 108 ac 64 m3 n2 Sometimes it is convenient to multiply a root by another quantity, or one root by another. If it is required to multiply (3 a b) by a b, it may be expressed thus: a b (3 ab). But if it is required actually to unite them, a b must first be raised to the second power, and the pro duct becomes (3 ab3). This will appear more plain in the following manner, (3 a2b)* = 3¥ This multiplied by a b is 3+ ab xab= 32 a2 b1a = 3a a2 ba = (3 aa b3)2. If instead of enclosing the quantity in the parenthesis and writing the exponent of the root over it, we divide the exponent of all the factors by the exponent of the root, all the operations will be very simple. That is, multiplication is performed on similar quantities by adding the exponents, as when the exponents are whole numbers. In like manner division is performed by subtracting the exponents. It must be observed that a3 may be read, the third root of the second power of a, or the second power of the third root of For That is, a power of a root may be found by multiplying the fractional exponent by the exponent of the power. Consequently a root of a root may be found by dividing the fractional index by the exponent of the root. In multiplying and dividing the fractional exponents, we must apply the same rules that we apply to common fractions. If the numerator and denominator both be multiplied or divided by the same number, the value of the quantity will not be altered; for that is the same as raising it to a power, and then extracting the root. If it is required to multiply a by a, the fractions may be reduced to a common denominator and added: thus, The same may be done in division and the exponents subtracted. The In fact, quantities with fractional exponents are subject to precisely the same rules, as when the exponents are whole numbers; but the rules must be applied as to fractions. fractions may be reduced to decimals without altering the value; thus .2 at = a1t = a1.25 = a × a·25 = a × a13× g-05 a2x a32 = a1.75 × a•6 = a .6 2.35 - 2 Χα = a2x Χ από κατάσ It is very important to remember how these quantities may be separated into factors. Since multiplication is performed by adding the exponents, and division by subtracting them, any quantity may be separated into as many factors as we please, by separating the exponent into parts. Thus,, a3 a3 X a2 = a × a1 = a × a2 × a3 = a3×a×a2 × a2 × a × aa × a3. Χα χα χ X X The sum of all the exponents in the last expression is 5. Logarithms are of the same nature as these exponents, and afford as great a facility in operating upon numbers, as these do upon letters. And the operations are performed in the same way, as will be explained hereafter. If the learner should ever have occasion to read other treatises on mathematics, he will generally find the roots expressed by what are called radical signs. The second root is expressed with the sign, the third root the same sign with the index of the root over it. The 4th root is, &c. 3 They will be easily understood if the radical sign be removed, and the exponents divided by the index of the root or the quan tity enclosed in a parenthesis, and the root written over it. The expression 5 a 3 becomes 4 The expression √ aa + b2 is equivalent to (a2 +6°)* XLI. Binomial Theorem. It has already been remarked that the powers of any quantity are found by multiplying the quantity into itself as many times, less one, as is expressed by the exponent of the power. Sir Isaac Newton discovered a method, by which any quantity consisting of more than one term may be raised to any power whatever, without going through the process of multiplication. The principle on which this method is founded is called the Binomial Theorem. Its use is very important and extensive in algebraic operations. Next to quantities consisting of only one term, binomials, or quantities consisting of two terms, are the most simple. Let a few of the powers of a + be found and their formation attended to. a* + 3 a3 x + 3 a2x2 + ax3 a3 x + 3 a2x2 + 3 α x3 + x2 (a + x)'= a + 4 a3 x + 6 a2 x2 +4α x2 + x a + x a+4 a x + 6 a3 x2 + 4 a2 x3 + α x2 a1 x + 4 α3 x2 + 6 α2 x 3 + 4 α x2 + 205 (a + x)3 = a3 + 5 a1 x + 10 a3 x2 + 10 a2 x3 + 5 a x1 + x® |