that the square of the greater divided by the less may be a minimum. Ans. The minimum required is 4d. and the value of the greater part corresponding is 2 d. 3. Let a and b be two numbers of which a is the greater, to find a number such, that if a be added to this number and b be subtracted from it, the product of the sum and difference thus obtained being divided by the square of the number; the quotient will be a maximum. 2 ab a b (a + b) a 4 ab 4. To divide a number 2 a into two parts such, that the sum of the quotients obtained by dividing the parts mutually one by the other may be a minimum. Ans. The number should be divided into two equal parts, and the minimum is 2. 5. To find a number such, that if a and b be added to this number respectively, the product of the two sums thus obtained divided by the number may be a minimum. Ans. The number a b, and the minimum, =(√a + √b)2 6. Let d be the difference between two squares; required the least value of the excess of the product of the greater of the two roots by a given number a above the product of the other root by a given number b, a being greater than b. Let x be the less root, the other will be ✔d +x2 and the minimum sought will be m =√ d (a2 — b2). 7. Let p and q be two given numbers, it is required to find SECTION XV.-POWERS AND ROOTS OF MONOMIALS. 123. When a quantity is multiplied into itself the product, we have seen, is called a power, the degree of which is marked by the exponent of the product, thus aaaaa or a3 is called the fifth power of a; in like manner am is called the mth power of a. The original quantity, from which a power is derived, is called the root of this power. The degree of the root is determined by the number of times the root is found as a factor in the power; thus a is the fifth root of a5; in like manner a is the mth root of am The number which marks the degree of the root is called the index of the root. 124. Let it be proposed to find the fifth power of 2 a3 b2; this power is indicated thus, (2 a3 b2)5, and we have, it is evident, (2 a3 b2)52 a3 b2 × 2 a3 b2 × 2 a3 b2 × 2 a3 b2 × 2 a3 b2 Here, it is evident, 1°. that the coefficient 2 must be multiplied into itself four times or raised to the fifth power, 2o. that each one of the exponents of the letters must be added, until it is taken as many times as there are units in the exponent of the power, or in other words, multiplied by 5; we have therefore (2 a3 b2)532 als blo In like manner (8 a2 b3 c)=512 a b c3. To raise a monomial therefore to any given power, we raise the coefficient to this power, and multiply each one of the exponents of the letters by the exponent of the power. With respect to the sign, with which the powers of a monomial should be affected, it is evident, that whatever be the sign of the quantity itself, its second power will be positive. Moreover if the exponent of the power of a monomial be an even number, it is easy to see, that this power may be considered as a power of the square of the proposed quantity. Thus a3, it is evident, may be considered the fourth power of a2; in like manner a2 m , any even power of a, may be considered the mth power of a2. It follows therefore, that whatever be the sign of a monomial, any power of it, the exponent of which is an even number, is positive. Again, since the power of a simple quantity, the exponent of which is an odd number, is equal to a power of this quantity of an even degree multiplied by the first power, it follows, that every power of a monomial, the exponent of which is an odd number, will have the same sign as the quantity from which it is ormed. 6 125. Let it now be proposed to find the third root of 64 a b3 c3. The root required is indicated, thus, 3√64 a b3 c3 ; the sign being employed to denote in general that a root is to be taken, and the index 3 placed above the radical sign to denote the particular root required. Since the root of a quantity must evidently be sought by a process the reverse of that, by which it is raised to a power, in order to extract the root of a monomial, 1°. we extract the root of the coefficient. 2°. we divide the exponent of each of the letters by the index of the root. According to this rule the third root of the proposed will be 4 a2 b3 c. In like manner the seventh root of a14 21 c56 is a2 b3 c3. With respect to the signs, with which the roots of monomials should be affected, it is an evident consequence of the principles already established that, 1o. Every root of an even degree of a have indifferently either the sign + or —. of 64 a18 is ± 2 a3. positive monomial may Thus the sixth root 2o. Every root, the degree of which is expressed by an odd number, will have the same sign as the quantity proposed. Thus the fifth root of 32 a10 65 is 10 -2 a2 b. 3o. Every root of an even degree of a negative monomial is an impossible or imaginary root. For there is no quantity, which raised to a power of an even degree can give a negative result. Thus a, b denote impossible or imaginary quantities, in the same manner as 4 6 a, 126. From what has been said, it is evident in order that a root may be extracted, 1o. that the coefficient of the proposed must be a perfect power of the degree marked by the index of the root to be extracted. 2°. that the exponents of each of the letters must be divisible by the index of the root. When this is not the case the root can only be indicated. It should be observed, however, that radical expressions, of any degree whatever admit of the same simplifications as those of the second degree. These simplifications are founded upon the principle, that any root whatever of a product is equal to the product of the same root of the several factors. Thus let it be proposed to find the third root of 54 a4 b3 c2. The third root, it is evident, cannot be taken; for 54 is not a perfect third power, and the exponents of the letters a and c are not divisible by 3. We therefore indicate the root, thus, 354 at 63 c2; but this expression may be put under the form 3√27 a3 b3 × 2 a c2; whence taking the third root of the factor 27 a3 63, we have 354 a4 63 c2-3 ab3/2 ac2. Let it be proposed next to find the third root of 125 a5 b+375 a3 c. This expression, which it is easy to see is not a perfect third power, may be put under the form 125 a3 (a2b+3c); whence extracting the third root of the first factor, we have for the root sought 5 a va2b+3c. EXAMPLES. 1. To reduce *80 a b c to its most simple form. 5 7 2. To reduce 128 x y z2 to its most simple form. ax + bx to its most simple form. 27 a3 +81 ab to its most simple form. 48 a 16 a be 81 a to its most simple form. SECTION XVI. POWERS OF COMPOUND QUANTITIES, THEORY OF COMBINATIONS, BINOMIAL THEOREM. Powers of compound quantities are found like those of monomials by the continued multiplication of the quantity into itself. They are indicated by inclosing the quantity in a parenthesis, to which is annexed the exponent of the power. The third power of a2+5 a bb c2, for example is indicated, thus, (a2+5abbc2)3. This same power may also be indicated thus, a2+5 a b — b c23 Next to monomials binomials are those, which are the least complicated. We begin therefore with these. Below are several of the first powers of the binomial x + a, (x + α)1 = x2 + 4 a x3 + 6 a2 x2 + 4 a3 x + aa We have formed the different powers of x+a in this table by the continued multiplication of x+a into itself. In this way we arrive only at particular results. To form any of the higher powers, the process of multiplication must still be continued. This would be tedious especially as the power, to which the binomial is to be raised, becomes more and more elevated. We proceed therefore to investigate a method, by which a binomial may be raised to any power whatever without the necessity of forming the inferior powers. This method was discovered by Newton. The principle on which it is founded is called the Binomial Theorem. The most simple and elementary demonstration of this theorem depends upon the theory of combinations, to which we shall first attend. THEORY OF COMBINATIONS. 127. The results, obtained by writing one after the other in every possible way a given number of letters, in such a manner, that all the letters will enter into each result, are called permutations. Let there be, for example, two letters, a and b. These give, it is evident, two permutations, a b, b a. Again, let there be three letters a, b, and c. If we set apart one of the letters, a for example, the remaining letters give two permutations, viz. b c, c b; placing next the a at the right of each of these, we have two permutations of three letters, viz. b c a, c ba; but each of the remaining letters b and c, being set apart in the same manner, will also furnish each two permutations of three letters; whence the permutations of three letters will be equal to the permutations of two letters multiplied by three. In like manner the permutations of four letters will be found equal to the permutations of three letters multiplied by four. |