✔, 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, In order to demonstrate this principle, let us remark, that, according to our definition of the square root of any expression, we have, Hence, since the squares of the quantities √a b c d ---- and vā. Vī √c. √d--- are equal, the quantities themselves must be equal. 4 This being established, the expression given above √98 a b 1 may be put under the form √ 49 ba × √ 2 a, but √ 49 ba is by (Art. 49) = 7 62; hence So also, √864 a b c = √144 a 10 × 6 b c = √144 a2b 4 10 11 2 4 с × √6bc = 12 a b c 5 √6 b 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 quantities 7 b, 3 a b c, 12 5 √2 a, 3 a b c √ 5 b d, 12 a b2 c3 √ 6 b c, &c. the 2 a b c3, 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 a2 b3, or of 5 a 2 b3, is + 25 a 4 b 6. Hence we conclude, that if a monomial be positive its square root may be either positive or negative. Thus, √9 a + = + 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 a1 = + 3 a 2, √ 144 a 2 b + c ® = + 12 a b 2 c 3. 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,√—Aα " 4 6 4 - to execute. √ — 5, are algebraic symbols which represent operations which it is impossible 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 a 3 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) 6, 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 b 2 ) 5 = 3 2 2 2 a3 b2 × 2 a 3 b 2 × 2 a 3 b 2 × 2 a 3 b2 × 2 a3 b2 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, 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 arithmetic. 2. Divide the exponent of each letter by the index of the required root. Thus, V/64 ab3 c 9 3 6 = 4 a 3 b c 2 129 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, \/ ́ before the quantity, and writing within it the index of the root. Thus, if it be required to extract the cube root of 4 a 265, the operation will be indicated by writing the expression, 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 nth roots of the different factors. Or, in algebraic language, = Raise each of these expressions to the power of n, then (Vabcd-)= Again, = abcd-.. Hence, since the nth powers of the quantities Vabcd, and Va.Vb.Vc. √π----, are equal, the quantities themselves must be equal. 3 This being established, let us take the expression † 54 a 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. We have, 3 (1.) /54 a1 b3 c2 = 3/27 × 2 × a3 × a × b 3 × c 2 = √27 × Vπ3 × √ b3 x V 2 ac2 by the principle just proved, In the above expressions, the quantities 3 a b, 2 a b2 c, 2 ac2, placed before the radical sign, are called the coefficients of the radical. 58. There is another principle which can frequently be employed with advantage in treating these quantities; this is, The mh power of the nth power of any quantity is equal to the m n1 power of that quantity. Or, in algebraic language, The 'mn' root of any quantity is equal to the mth root of the nth root of that quantity. Or, in algebraic language, Hence, as often as the index of the root is a number composed of two or more factors, we may obtain the root required by extracting in succession the roots whose indices are the factors of that number. Thus, That is to say, that when the index of the radical is multiplied by a certain number n, and the quantity under the radical sign is an exact n1h power, we can, without changing the value of the radical, divide its index by n, and extract the nth root of the quantity under the sign. 59. This last proposition is the converse of another not less important, which consists in this, that we may multiply the index of a radical by any number, provided we raise the quantity under the sign to the power whose degree is marked by that number, or, in algebraic language, For in fact, a is the same thing as "VaTM, and therefore, 60. By aid of this last principle, we can always reduce two or more radicals of different degrees to others which shall have the same index. Let it be required, for example, to reduce the two radicals / and /3bc to others which shall be equivalent, and have the same index. If we multiply 3 the index of the first, by 5 the index of the second, and at the same time, raise 2 a to the 5th power; if, in like manner, we multiply 5 the index of the second, by 3 the index of the first, and at the same time raise 3 b c to the 3d power, we shall not change the value of the two radicals, which will thus become In order to reduce two or more radicals to others which shall be equivalent and have the same index, multiply the index of each radical by the product of the indices of all the others, and raise the quantity under the sign to the power whose degree is marked by that product. 2 3 5 Thus, let it be required to reduce √/2a, √/3b * c 3, {/⁄ 4 d a e 3 ƒ“, to the same index, The above rule, which bears a great analogy to that given for the reduction of fractions to a common denominator, is susceptible of the same modifications. Let it be required, for example, to reduce the radicals, V, V5b, VTC, to the same index: since the least common multiple of the numbers 4, 6, 8, is 24, it will be sufficient to multiply the index of the first by 6, of the second by 4, and of the third by 3, raising the quantities under the radical in each case to the powers of 6, 4, 3, respectively, 3 Va = Vā, V56 = 625, V3c = V27 c* |
