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, if the last rule be applied to the second of these quantities, it will produce the first. 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 V2a and V36c 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 2a 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 3bc to the 3d power, we shall not change the value of the two radicals, which will thus become =V32a5 5X We shall thus have the following general RULE. 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. Thus, let it be required to reduce √2a, ¥/3b°c3, V/4d'ef to the same index, √2a =216a15 =2×5× √(3b2c3) 2×5 = 3031062030 √ 4d1e3ƒ©=2×3× {√ (4d+e$ƒ©)2×3 = W 4®d2e30ƒ 36 ̧ 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 Ya, 5b, 2c 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, (1) Reduce VaTM, (2) Reduce a, (3) Reduce Va3, a=a, 5b-625b, 3c= 27c. EXAMPLES. ", and c to the same index. 61. Let us now proceed to execute upon radicals the fundamental operations of arithmetic. ADDITION AND SUBTRACTION OF RADICALS. DEFINITION.-Radicals are said to be similar when they have the same index, and when, also, the quantity under the radical sign is the same in each; thus, 3 √a, 12ac√a, 15b √a, are similar radicals, as are, also, 4ab√mn2p3, 51mn p3, 25dymn2p3, &c. This being premised, in order to add or subtract two similar radicals we have the following RULE. Add or subtract their coefficients, and place the sum or difference as a coefficient before the common radical. For example, (1) 337+2y=53/b. (2) 346-24T=yb. (3) 3pqmn+41√mn=(3pq+41) ☆ mn.* (4) 9cd va-4cd √a=5cd √a. If the radicals are not similar, we can only indicate the addition or subtraction by interposing the signs + or It frequently happens that two radicals, which do not at first appear similar, may become so by simplification; thus, (5) √48ab2+b√75a=√3×16×a× b2 + b √ 3 × 25 × a =4b√3a+5b√3a (6) 2√45—3√5=2√5×9—3√5 =3√5. (7) V/8a3b+16u-/b*+2ab3/8a3(b+2a)—Yb3(b+2a) =(2a—b)2a+b. * When two products, consisting each of several factors, have any common factors, the other factors may be regarded as the coefficients of these, since they show how many times the common factors are repeated, and the addition may be performed by adding the coefficients, and annexing the common factors to the sum; thus, abcd+mned=(ab+mn)cd, and 5ab√x+4cb√x=(5a+4c)b√x, on the same principle as 8a+4a=12a. (8) 3/4a2+2/2a=34/2a+22/2a =5/2a. (9) √8+ √50— √18=4√2. (10) bab+cac-dad (b2 - c2-d2) a. (11) 2√√60-√15+ √ √15.* MULTIPLICATION AND DIVISION OF RADICALS. 62. In the first place, with regard to radicals which have the same index, let it be required to multiply or divide Va by V, then we shall have Vax Vb=Vab, and Va÷Võ=√4. For, if we raise Vax √b, and Vab, each to the nth power, we obtain the same result, ab; hence these two expressions are equal. The same principle is demonstrated in (57). In like manner, Va and n Vb b' a when raised to the nth power, give; hence the two expressions are equal. We shall thus have the following RULE. In order to multiply or divide two radicals which have the same index, multiply or divide the quantities under the sign by each other, and affect the result with the common radical sign. If there be any coefficients, we commence by multiplying or dividing them separately. The latter part of this rule depends upon the principles set forth and alluded to in 17, note; the coefficients, or rational parts, and the radical parts being regarded as factors composing a product. (a2 + b2) 3 cd (1) 2a 3f(a 23 (a2+b2). X - 3a C (a2 + b2)2 -6a2 * The numerator and denominator of each of the two fractions in this example are multiplied by its denominator. The denominator becomes thus a perfect square, and may be set outside the radical sign. (17) (√15+√√—12— √—21)÷√−3=2+ √5—√7. If the radicals have not the same index, we must reduce them to others having the same index, and then operate upon them as above; thus, (1) 3a bx5b2c=3a b'x5b8c FORMATION OF POWERS AND EXTRACTION OF ROOTS OF RADICALS. 63. Let it be required to raise a to the nth power; then, =Va", according to the rule for multiplication just established. Hence we have the following RULE. In order to raise a radical quantity to any given power, raise the quantity under the sign to that power, and place over the result the radical sign with its original index. If there be any coefficient, we must raise the coefficient separately to the required power. Thus, When the index of the radical is a multiple of the exponent of the power which we wish to form, the operation may be simplified. Let it be required, for example, to square 2a; we have seen (Art. 58) that √√√√a; but in order to square this quantity, it is sufficient to sup press the first radical sign; hence, (√2a)2 = √2a. Again, let it be required abc to the 5th power; now, Wabc=√√√abc; but in order to raise this quantity to the 5th power, it is sufficient to suppress the first radical sign; hence, (abc)=√abc, and, in general, to raise If the index of the radical be divisible by the index of the required power, we may divide the index of the radical by the index of the power, and leave the quantity under the sign unchanged. 64. With regard to the extraction of roots, either by virtue of the principle established in (Art. 59), or by reversing the last rule, we shall manifestly have the following RULE. In order to extract any root of a radical quantity, multiply the index of the radical by the index of the root required, and leave the quantity under the sign unchanged. If there be a coefficient, we must extract its root separately. Thus, (1) √√√3c=43c. (2) √√√√5a=√5a. (3) √803ab=2cab. If the quantity under the sign be a perfect power of the same degree as the root required, we may simplify. Thus, *It may be well to note here that the even power of a radical of the second degree is rational, and the uneven power irrational, the latter being formed by the multiplication of the proposed radical by a rational quantity. |