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ON THE FORMATION OF POWERS, AND THE EXRACTION OF ROOTS OF ALGEBRAIC
48. We begin by considering the case of monomials, and, in order to simplify the subject as much as possible, we shall first treat of the formation of the square and the extraction of the square root only, and then proceed to generalize our reasonings in such a manner as to embrace powers and roots of any degree whatsoever.
DEFINITION. The square root of any expression is that quantity which, when multiplied by itself, will produce the proposed expression. Thus the square root of a2 is a, because a, when multiplied by itself, produces a 2; the square root of (a + b) is a + b, because a + b, when multiplied by itself, produces (a+b); in like manner 8 is the square root of 64, 12 of 144, and so on. The process of finding the square root of any quantity is called the extraction of the
The extraction of the square root is indicated by prefixing the symbol V to the quantity whose root is required. Thus a signifies that the square root of a is to be extracted; √ a 2 + 2 a b + b2 signifies that the square root of a 2+2 a b + b2 is to be extracted, &c.
In order to discover the method which we must pursue in order to extract the square root of a monomial, let us consider in what manner we form its square. According to the rule for the multiplication of monomials,
=5a2bc X 5 a 2 Z 3 c
(9 a b2 c3 d 4) 2 And, (Army ---) Ar" y "z"---X Ax" y "z"--- A2x2 my in2th..
= 9 ab с 3d4 X 9ab2 c3 d4 81 a 2 b c do
49. Hence it appears, that, in order to square a monomial, we must square its coefficient, and multiply the exponents of each of the different letters by 2. Therefore, in order to derive the square root of a monomial from its square, we must,
L Extract the square root of its coefficient according to the rules of Arith metic.
II. Divide each of the exponents by 2.
Thus we shall have,
√625 a*b*c Here also,
This is manifestly the true result, for
√64 a 6 b = 8a3b2
=25 a b c
= 8 a3b* X 8 a b2 = 64 a 6 b1
50. It appears from the preceding rule, that a monomial cannot be the square of another monomial unless its coefficient be a square number, and the exponents of the different letters all even numbers. Thus 98 a b is not a perfect square, for 18 is not a square number, and the exponent of a is not an even number. In this case we introduce the quantity into our calculations, affected with the sign
EXTRACTION OF ROOTS.
/, and it is written under the form √98 a b 4. Expressions of this nature are called Surds, or Radicals, of the Second degree.
51. Such expressions can frequently be simplified by the application of the following principle :--: -The square root of the product of two or more factors is equal to the product of the square roots of these factors. Or, in algebraic language,
và x Vô x và x và
In order to demonstrate this principle, let us remark, that, according to our definition of the square root of any expression, we have,
√98 b * a
Hence, since the squares of the quantities va b c d -.. and vā. Vī vo. Vd--- are equal, the quantities themselves must be equal. This being established, the expression given above √ 98 a b 1 may be put under the form 49 b* × √2 a, but √ 49 b1 is by (Art. 49) = 7 b2; hence
= √ 49 b2 × √ Ta
So also, √r6+ a2b3 c = √]
√ 45 a*b*c* d = √9a*b*c* × 5 b ȧ
= √9a*b*c* X √5bd
C × 6 b c = √144 a * b a ̈ с 10 x vobi
c √бb c
In general, therefore, in order to simplify a monomial radical of the second degree, separate those factors which are perfect squares, extract their root (Art. 49), place the product of all these roots before the radical sign, under which are to be included all those factors which are not perfect squares.
In the expressions, 7 b 2 √ 2 a, 3 a b c √ 5 b d, 12 a b2 c3 √6 b c, &c. the quantities 7 b', 3 a b c, 12 a b 2 c 3, are called the coefficients of the radical.
52. We have not hitherto considered the sign with which the radical may be affected. But since, as will be seen hereafter, in the solution of problems we are led to consider monomials affected with the sign —, as well as the sign +, it is necessary that we should know how to treat such quantities. Now the square of a monomial being the product of the monomial by itself, it necessarily follows, that whatever may be the sign of a monomial, its square must be affected with the sign +. Thus, the square of +5 a b3, or of 5 a 2 b3, is +25 ab 6.
Hence we conclude, that if a monomial be positive its square root may be either positive or negative. Thus, 9 a * = 4 + 3 a 2, or - 3 a 2, for either of these quantities, when multiplied by itself, produces 9 a*; we therefore always affect the square root of a quantity with the double sign+, which is called plus or minus. Thus, √9 a 1 = + 3 a 2, √ 144 a 2 b 1 c ® = + 12 a b 2 c3.
53. If the monomial be affected with a negative sign, the extraction of its square root is impossible, since we have just seen that the square of every quantity, whether positive or negative, is essentially positive. Thus, √9, √ a
√ — 5, are algebraic symbols which represent operations which it is impossible to execute. Quantities of this nature are called imaginary, or, impossible, quantities, and are symbols of absurdity which we frequently meet with in resolving quadratic equations.
By an extension of our principles, however, we perform the same operations upon quantities of this nature as upon ordinary surds. Thus, by (Art. 51).
54. Let us now proceed to consider the formation of powers and extraction of roots of any degree in monomial algebraic quantities.
DEFINITION. The cube root of any expression is that quantity which, multiplied twice by itself, will produce the proposed expression. The fourth, or, biquadrate root of any expression is that quantity which, multiplied three times by itself, will produce the proposed expression; and in general, the n' root of any expression is that quantity which, multiplied (n − 1) times by itself, will produce the proposed expression. Thus, the cube root of a3 b3 is a b, because a b, multiplied by itself twice, produces a3b3; for the same reason, (a + b) is the 6th root of (a+b), 2 is the seventh root of 128, &c. .
55. Let it be required to form the fifth power of 2 a 3 b2.
(2 a3 b2) 5 =
(2 a b2 c3 d1)"
Where we perceive, 1°. That the coefficient has been raised to the fifth power; 2°. That the exponent of each of the letters has been multiplied by 5. In like manner,
(8 a 2 b3 c) 3 =
2 a3 b2 × 2 a3 b2 × 2 a3 b2 × 2 a 3 b 2 × 2 a 3 b 2
8 a 2 b3 c X 8 a 2 b 3 c × 8 a 2 b3 c
2 ab2 c3 d1× 2 a b 2 c3 d1×
Hence we deduce the following general
RULE TO RAISE A MONOMIAL TO ANY POWER.
Raise the numerical coefficient to the given power, and multiply the exponents of each of the letters by the index of the power required. And hence reciprocally we obtain a
RULE TO EXTRACT THE ROOT, OF ANY DEGREE, OF A MONOMIAL
1o. Extract the root of the numerical coefficient according to the rules of arith
2. Divide the exponent of each letter by the index of the required root.
64 a 9 Z3 c6
= 4 a b c2
56 According to this rule, we perceive that in order that a monomial may be a perfect power of that degree whose root is required, its coefficient must be a perfect power of that degree, and the exponent of each letter must be divisible by the index of the root.
When the monomial whose root is required is not a perfect power of the required degree, we can only indicate the operation by placing the sign, 1 ~ before the quantity, and writing within it the index of the root. Thus, if it be required to extract the cube root of 4 ab, the operation will be indicated by writing the expression,
2 a2b3c4 d.
Expressions of this nature are called surds, or, irrational quantities, or, radicals of the second, third, or, nth degree, according to the index of the root required.
57. We can frequently simplify these quantities by the application of the following principle, which is merely an extension of that already proved in (Art. 51).
The nth root of the product of any number of factors is equal to the product of the n' roots of the different fuctors. Or, in algebraic language,
= √ā× võ× Võ × √ π × ---Raise each of these expressions to the power of n, then (Vabcd-----) = a b c d -
Hence, since the n powers of the quantities Vabcd, and Va. Vb. Vč. √☎----, are equal, the quantities themselves must be equal.
This being established, let us take the expression √ 54 a1 b3 c2, whose root cannot be exactly extracted, since 54 is not a perfect cube, and the exponents of a and c are not exactly divisible by 3.
(1.) /54 a*b*c* = /27 × 2 × a3 xa x b3 x c2 = √2 × Va3 × Vo3× V2 ac2
by the principle just proved,
(2.) 1/48 a b c = 16 x 3 x a1 x a x b® X c1 X c1 = √16× Vā1× Võ 3× V ca × V 3 × Vā × Vē 8X = 2ab2c3ac