is 4; 2 is the cube root of 8, the fourth root of 16, the fifth root of 32, and so on.

451. The sign is termed the radical sign; when placed before any number, either by itself, thus 4, or with a small 2 annexed, 24, it signifies the square root of that number. Thus, the above represents the square root of 4, which is 2; therefore 4 is the square, and 2 the square root. 3/8 represents the cube root of 8; therefore 8 is the cube and 2 is the cube root. 16 represents the fourth root of 16. 5/32 represents the fifth root of 32, and so on.

452. The root of a number is frequently denoted by a fractional exponent; 4* denotes the square root of 4. 8* denotes the cube root of 8, &c. Thus,

4* = √/ 4 = 2 = √/ 2.2 = (2.2)a &c. 83 = 3√/8 = 2 =

2.2.2.(2.2.2.).

3

8 represents the cube root of the square of 8. The square of 8 is 64, and the cube root of 64 is 4, for 4 × 4 × 464; but 8 also represents the square of the cube root of 8. The cube root of 8 is 2, and the square of 2 is 4.

3

Thus 83 (8 = (8. 8) 3 = (64) § = 3√√/64 = 3√√/8.8=4

=22 = 2 × 2 = 4.

453. From the above statement, the student will perceive that when both a power and a root of a number are intended to be represented, it is done by annexing a small fraction to the number, the numerator of that fraction representing the power to which the number is raised, and the denominator representing the root to be extracted.

454. The different roots of a number may be multiplied together or divided by adding or subtracting their fractional exponents. To add or subtract fractions, the student will recollect we must first reduce them to a cominon denominator. To multiply the square root of 61 by the cube root of 64, is to add their fractional exponents. Therefore 64 × 3√/64 = 61a × 643 = 64a × 643 = 64%+3=64a="®√/64* = (64°); = (643)2 = 8×4 = 32.

5

5

3

455. To divide one root of a number by another root of that number, we subtract their fractional exponents. Therefore

455. Powers of a number and roots of a number may also be multiplied together, or the one divided by the other, by adding

or subtracting their exponents, as 42 X 4*

1. What is understood by the powers of a number? 2. What are exponents?

=

= 41-4

3. Is the adding of exponents equivalent to the multiplication of powers? Is the subtraction of exponents equivalent to the division of powers?

4. What is understood by the negative powers of a number?

5. Is the effect of negative exponents contrary to that of positive ones?

6. What is the effect of multiplying a negative and a posi tive power together, whose exponents are the same?

7. What is the effect of dividing a negative power by a positive whose exponents are the same?

8. What is a number called by whose continued product a power is produced? That is, if a number be multiplied a certain number of times into itself, what is such number called with reference to its product?

9. What is the radical sign?

10. How do you represent the square root of a number? 11. How do you represent the cube root of a number?

12. How do you represent the 2nd, 3rd, and 4th powers of a number?

13. When both the power and root of a number are to be represented, how is it done? For instance, the square root of the third power of 6 ?

179

CHAPTER XVII.

ALGEBRA.

457. No branch of mathematical science is more important, or of more extensive utility, than Algebra. So extensive are the principles which it embraces, that every rule, every principle in arithmetic is founded on algebraic deduction. If Algebra were more generally understood, we should have more expert arithmeticians. Attention, however, is now beginning to be paid in schools to this department of knowledge, and it is to be hoped that in a few years it will become a common branch of education.

458. There can be no reason why the study of Algebra should not be connected with the study of common arithmetic ; there are, on the contrary, many reasons why it should; for while, on the one hand, the notation of Algebra is eminently useful in explaining and shortening the operations of arithmetic, the operations of arithmetic, on the other, supply the clearest illustrations of the results of Algebra.

459. It is not my object to write a treatise on Algebra, but merely to give the student such an introduction to the science as will enable him to understand its elementary principles, the understanding of which will greatly assist him to a full and clear conception of the rules which are to follow. Such an introduction, and such an understanding on the part of the pupil, will also enable me to explain with much greater conciseness the principles of the succeeding rules in arithmetic, than I should otherwise be enabled to do. Another advantage will arise from this connexion of Algebra with arithmetical knowledge; it will accustom the student to the language and terms of a new science, and by giving him an insight into its principles will enable him with much greater ease and pleasure to enter at some future time more deeply into the study of this important branch of the higher mathematics.

The

460. ALGEBRA is arithmetic under another form. principles of both are the same, and their operations are very similar. Their notation, however, is different. In arithmetic the quantities on which we operate being always known, are always expressed in figures. In Algebra, the quantities on which we operate are frequently unknown or indefinite; such quantities, therefore, cannot be expressed in numbers.

461. To represent such quantities, the letters in the alphabet are used. The first letters of the alphabet, such as a, b, c, &c. are generally used to representknown quantities, and unknown quantities are generally represented by the last letters of the alphabet, as x, y, z. These symbols may represent any quantities at pleasure. If a = 12, b = 8, and c = 6, then a + b+c represents the sum of a + b + c = 12 + 8 + 6 = 26, a+b c represents the difference between the sum of a + b, and c, as 12+ 8 6 614. a X b xc, or a . b. c, represents that a, b, and c, have to be multiplied into each other, and their product is represented by a b c ; as a. b. c = abc57612 X 8 X 6. a.bc, or

=20

a.b

с

represents that

a and b have to be multiplied together and their product divided by c. abc or represents the quotient of a b di

ab

с

=16=

6

12.8

6

462. 2 a represents that the quantity expressed by a, is to to be taken twice. When numbers are placed before quantities in this manner, they are called the co-efficients of such quantities.

463. When the same quantity has to be multiplied into itself a certain number of times, the product is called a power of that quantity; thus, the product of a. a is represented by aa, or more generally by a". It is called the 2nd power or square of a. The product of a.a. a is represented by aaa or a. It is called the 3rd power or cube of a. The square root of a is represented by a or at. The cube root of a by 3 a or as. The small figures, which are placed above a quantity to the right, as exhibited above, and denoting the power or root of a quantity, are called the exponents or indices of the quantities to which they are affixed.

464. Quantities expressed by the same letter, are called like quantities. Quantities expressed by different letters, are called unlike quantities.

465. When quantities have the sign + before them, they are called positive quantities. When quantities have the sign before them, they are called negative quantities. When quantities have no sign before them, the sign + is understood.

466. When two quantities or collection of quantities are expressed with the sign between them, such an expression is called an equation; thus, 4 × 6 = 24, is an equation, as also is 6 X 4 - 4 = 5 × (6· 2) or ab+c=12b+c.

465. When operations are to be performed on compound quantities, to prevent mistakes such quantities are enclosed in brackets, or have a line drawn over them. - (b—c)

Thus a

or a-b-c, means that b diminished by c, is to be taken from a. Therefore as a 12 and b-c 2, a- -b-c = 10. Were the same expressed without brackets, thus, a-b-c, it would mean that b was to be taken from a, and then c from the remainder, or that both b and c were to be taken from a, which would give a result less than nothing; as 12—8—6—— 2, or 12— (8 + 6) = − -2. Negative results are not unusual in operations in Algebra.

ALGEBRAIC NOTATION.

466. It is stated above, that in Algebra the letters of the alphabet are used instead of numbers, to represent quantities, and that these quantities may be known or unknown. Now, when a letter is used to represent an unknown quantity, it supplies the place of that unknown quantity in all the reasonings and operations upon it, until it be known. Thus, if a = 12, b = 8, c = 4, and d = 3; and if x = a + b c, the quantities a, b, a nd c, being known, we easily find x, which is equal to 128. 4 16, These letters being substituted for numbers, are subject to all the operations of numbers. They are capable of being added, subtracted, multiplied, and divided, the same as numbers. Thus a+b represent, that the two numbers represented by a and b are to be added together. α- b represents the difference between a and b. a b represents that a and ¿ are to be multiplied together. And represents that a is to be divided by b. The following equations will show the notation of Algebra, compared with arithmetic :—

467. When an expression is enclosed in brackets, it is to be understood that the whole expression is to be considered as one quantity. Thus, by a + (b + c — d) we are to understand the quantity b, added to c, diminished by d, is to be added to a. Now, if we add b and c to a, we shall add too much, as it is not c and b that is to be added, but the sum of c and b diminished by d.

R