see that, when a fraction, having several terms in the numerator, is represented as subtracted, on removing the denominator, we change the signs of all the terms of the numerator. If, however, the fraction is preceded by the sign, no change is to be made in the signs of the numerator, on removing the denominator.

9. If from a certain number I subtract 10, and then subtract of this remainder from the original number, the last remainder will exceed of the original number by 6. Required the number.

10. A grocer had two casks full of wine, one containing twice as much as the other. From the smaller leaked out 13, and from the larger 46, gallons. He then drew from the smaller as many gallons as remained in the larger, and from the larger as many gallons as remained in the smaller; after which there remained in the larger 1 gallon more than in the smaller. How many gallons did each cask hold?

11. A and B are of the same age; but if A were 10 and B 5 years younger, of B's age subtracted from A's, would leave the same remainder as if of A's were subtracted from B's. Required their ages.

12. A man, having a lease for 100 years, on being asked how much of it had transpired, said, that § of the time past subtracted from the time to come, would leave the same remainder, as if of the time to come were subtracted from the time past. had transpired?

How many years of the lease

13. There is a certain number such, that if 25 be subtracted from it, and if it be subtracted from 125, g of the first remainder subtracted from of the second, will leave 10 more than of the original number. Required that number.

ART. 27. In pure algebra, letters are generally used to represent known as well as unknown quantities. We shall now treat of operations upon quantities purely algebraical.

It is to be observed, that the addition, subtraction, multiplication, &c. of algebraic quantities cannot, strictly speaking, be performed, in the same sense as they are in arithmetic, but are, in general, only represented; these representations, however, receive the same names as the actual operations in arithmetic.

A monomial is a quantity consisting of a single term; as,

A binomial is a quantity consisting of two terms; as, a+m, or x―y.

A trinomial is a quantity consisting of three terms; as, a+b-c.

Polynomial is a general name applied to any quantity containing several terms.

ART. 28. The product of monomials is expressed by writing them after each other, either with or without the sign of multiplication; thus, a. b, a × b, or ab, signifies that a and b are multiplied together. The last form, viz. ab, is generally used. In like manner, the product of a, b, c, and d, is a b c d.

The order of the letters in a product is unimportant; thus, ab is the same as ba. This will be manifest, if

SALEM FRATERNITY.

87.]

MULTIPLICATION OF MONOMIALS.

51

numbers are put instead of the letters. Suppose a = 4, and b=5; then a b=4.5=20, and ba=5.4=20.

Hence, each of the expressions, abc, a cb, bca, bac, cba, and cab, is the product of a, b, and c. For the sake of uniformity, however, the letters of a product are usually written in alphabetical order.

If 3 ab and 5 m n were to be multiplied together, we might write the quantities thus, 3 ab 5 mn; but, since the order of the factors is unimportant, the numerical factors may be placed next to each other; thus, 3.5 abmn. Now, performing the multiplication of 3 by 5, we have 15 abm n.

But it would be erroneous to write 35 abmn as the product of 3 ab and 5mn, because the value of a figuré depends on its place with regard to other figures. If we would represent the multiplication of the figures also, they must be separated either by letters or by the sign of multiplication; as, 3ab5mn, 3.5 abmn, or 3 X5 abmn.

In like manner, the product of 2 am, 3xy, and 5 cd, is 30 a cdmxy; that of 7, 2b, and 4 d, is 56 b d.

We infer, from the preceding examples, that the product of several monomials consists of the product of the co efficients and all the letters of the several quantities.

1. Multiply 2 cd by 3 ax.

2. Multiply 3pq by 7.
3. Multiply 4 a by 3b.
4. Multiply 17 c by 2 m n.
5. Multiply 5 ax by 11 y.
6. Multiply 7 pq by 4rs.
7. Multiply 9gh by 11 x.
8. Multiply 20 by 3 x y.
9. Multiply 13 m x by 7 z.
10. Multiply 2 ef by 5x.
H. Multiply 25 by 8 abc.
12. Multiply mm by mmm.

In the last example, the product is, according to what has been shown above, mm mmm. But when the same letter occurs several times as a factor in any quantity, instead of writing that letter so many times, it is usual to write it once only, and place a small figure, a little elevated, at the right, to show how many times that letter is a factor. Thus, instead of mm mmm, we write m5. In the same manner, we write a6, instead of a aa aaa.

In all cases, a product contains all the factors of both multiplicand and multiplier. In the 12th example, m is twice a factor in the multiplicand, and three times in the multiplier; the product, therefore, must contain it five times as a factor; that is, m2 multiplied by m3, gives m5. The product of 2 a2b3 and 3 a4 b4 is 6ab7, because each letter must be found as many times a factor in the product as it is in both multiplicand and multiplier.

ART. 29. The small figure, placed at the right of a letter, is called the index or exponent of that letter, and affects that letter only, after which it is immediately placed. An exponent, then, shows how many times a quantity is used as a factor.

Quantities written with exponents are called powers of those quantities; thus, m2 is called the second power of m, m3 the third power, m5 the fifth power, &c.; and when a quantity is written without any exponent, it is supposed to have 1 for its exponent, and is called the first power of that quantity; thus, m, which is the same as m1, is called the first power of m.

Sometimes the second power is called the square, and the third power the cube, of a quantity names which,. though derived from geometry, are, for the sake of conciseness, very convenient in algebra.

Thus, m2 is read m square, m3 is read m cube.

Exponents must be carefully distinguished from coefficients; for 4a and at are very different in their value. Suppose a to be 5; then 4 a 20, and a15.5.5.5= 625.

=

ART. 30. From what precedes, we derive the following

RULE FOR THE MULTIPLICATION OF MONOMIALS.

Multiply the coefficients together, and write after their product all the different letters of the several quantities, giving to each letter an exponent equal to the sum of its exponents in all the quantities.

1. Multiply 3 a2m by 4a3m5.

In this question, the product of the coefficients is 12; the sum of the exponents of a is 5, and the sum of the exponents of m is 6. The answer, therefore, is 12 a3 mồ. 2. Multiply a2 b2 by 3ab3.

In this example, the product of the coefficients is 1.3 or 3; the sum of the exponents of a is 3, that of the exponents of b is 5; the answer, therefore, is 3 a3 b5.