If it is required to multiply (3 aa b) by ab, it may be expressed thus: a b (3 a2 b). But if it is required actually to unite them, ab must first be raised to the se cond power, and the product becomes (3 ab3). This will appear more plain in the following manner, 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. Let a be multiplied by a}. | X a = a3 + 3 = a. a3 × a3 = ai + i = aa2 = a2. a3 b3 × aš b} = aš +§ bš +3 = aš b7. 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 For the 3d root of a2 is a3; and the third root of a. a} x at = a} + ÷ = a}. } a3. The 3d power of a3 is a3 × a3 × a3 = a} + ÷ + } = a3} × = a3. 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. The 3d root of a is at. The 3d root of a is aš. The 5th root of a b3 is air bi‰. 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 to raising it to a power, and then extracting the root. If it is required to multiply a by a3, the fractions may be reduced to a common denominator and added: thus, αν και 7 að × a& = a2 = a1s = a.a3. = a X The same may be done in division and the exponents subtracted. 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. The fractions may be reduced to decimals without altering the value; thus 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 a = a3 × as = x = ax a2 × a3 = a αν καὶ καὶ καὶ καὶ καὶ και. χα 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. They will be easily understood if the radical sign be removed, and the exponents divided by the index of the root or the quantity enclosed in a parenthesis, and 4 the root written over it. The expression 5 a b3 be comes 5* 3 5ì a* b3 = (5 a2 b3)*. The expression a2 + b2 is equivalent to (a2 +b2)3. 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. |