3 a 1.75 ха 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 = al x a? =a x a* = a x a' x a? xat xarx at x at x aš xas. 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 o the same sign with the index of the root over it. The 4th root is -, &c. 5 3 a =ņa aš = va3 23 = ab 2 dob", &c. They will be easily understood if the radical sign be removed, and the exponents divided by the index of the ro or the quantity enclosed in a parenthesis, and the root written over it. The expression ✓ 5 a' b; becomes 57 aš v = (5 do 63). The expression Va® + * is equivalent to (a+62) XLI. 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 + x be found and their formation attended to. (a + x)' = a + x pommern a + 3 (a + x) = a* + 4 ao x + 6 d?m? + 4 ax* + ** a + x a® + 4. a* x + 6 ao x2 + 4 a* x + ax* a4 x + 4 a* x2 + 6 a' x3 + 4 a x4 + 20 (a + x) = a + 5 a* x + 10 a: x* +- 10 a' x3 + 5 ax* + . The law of the formation of the literal part is sufficiently manifest. In each power there is one term more than the number denoting the power to which it is raised. The first power consists of two terms, the second power of three terms, the third power of four terms, &c. In every power a is found in every term except the last, and x is found in every term except the first. The exponent of a in the first term is the same as the exponent of the power to which the binomial is raised, and it diminishes by one in each succeeding term. The exponent of x in the second term is 1, and it increases by one in each succeeding term, until in the last term it is the same as that of a in the first term. The law of the coefficients is not so simple, though it is not less remarkable. The coefficients of the first power, viz. a + x, are 1,1; those of the second power are 1, 2, 1. These are formed from the first as follows. When a is multiplied by a, it produces a', and no other term being produced like it, there is nothing added to it, and it remains with the same coefficient as the a in the multiplicand. In multiplying x by a and afterward a by x, two similar terms are produced, having the coefficients of the a and x in the multiplicand, viz. 1 and 1 ; and the addition of these forms the 2. The other 1 is produced like the first. The coefficients of the third power are 1, 3, 3, 1. The 1s are produced from the second power, as those of the second power are produced from the first. In multiplying 2 a x by a, the term produced is 2 ao , having the coefficient of the second term of the multiplicand; and in multiplying a' by x, the term produced is ao x, similar to the last, and having the coefficient 1 of the first term of the multiplicand. The addition of the coefficients of these two terms produces the 3 before a' x. That is, the coefficient of the second term of the third power is formed by adding together the coefficients of the first and second terms of the second power. In the same manner it may be shown, that the coefficient 3 of the third term of the third power is formed by adding together the coefficients of the second and third terms of the second power. The following law will be found on examination to be general. The coefficient of the first term of every power is 1. The coefficient of the second term of every power is formed by adding together the coefficients of the first and second terms of the preceding power. The coefficient of the third term of every power is formed by adding together the coefficients of the second and third terms of the preceding power. The coefficient of the fourth term of every power is found by adding together the coefficients of the third and fourth terms of the preceding power. And so of the rest. This law, though perhaps sufficiently evident by inspection, may be easily demonstrated. Suppose the above law to hold true as far as some power which we may designate by n. The literal part of the nth power will be formed thus. an, an-'x, ama mo, anam ......... a 2., 07. We cannot write all the terms without assigning a particular value to n. We can write a few of the first and last. The points between show that the number of terms is indeterminate ; there may or may not be more than are written. Suppose that A is the coefficient of the second term, B that of the third, &c. and let the whole be multiplied by a + , which will produce the next higher power, or the (n + 1)th power. |