To prove this, take the quantity Vabc, which is the nth root of abc, and we say that it is equal to the nth root of a, multiplied by the nth root of b, multiplied by the nth root of c; and that we shall have, Vabe Vax VbX Vc; for if we raise both quantities to their nth powers, we shall have The nth powers of these quantities being, therefore, equal, it is evident that the quantities themselves are equal. Now, in taking a quantity from under the radical sign, we extract the root of one of the factors of the radical; and since we multiply the coefficient of the radical by this root, the expression is still the product of the roots of the factors. Thus, Vab is the product of the square roots of a2 and b; and avb is evidently the same; the only difference is in the form of the expression. In the first, the square root of a2 is indicated by the sign, and in the last, it has been actually extracted. EXAMPLES. 1. Reduce 96a b'c to its simplest form. 3 3 V96a3b*c = √8a3b3× 12a2bc — 2ab12a2bc. 2. Reduce 243a2b5 to its simplest form. √243a2b3 = √⁄81b1×3a2b = 3b √3a2b. 3. Reduce √3a2b + 6ab2 +3b3 to its simplest form. √3a2b+6ab2+3b3= √(a2+2ab+b2)3b=(a+b)√3b. 3 4. Reduce V8ab2+24a3b to its simplest form. 3 3 3 √8a1b2+24a3b= √/8a3(ab2+3b)=2a√ab2+3b. 5. Reduce 54abc3 to its simplest form. Ans. 3bc Vbac. 6. Reduce 32a b c to its simplest form. Ans. 2ac V2a2b3. 7. Reduce 416a'c3b to its simplest form. 8. Reduce 2 to its simplest form. 3 Ans. 8ac V2ab. 2√2×3=2×1√3 = {√3, the ans. NOTE. If the quantity under the radical sign be a fraction, it can always be changed to a whole number, by multiplying both terms of the fraction by such a quantity as shall make the denominator a perfect power; thus, √} = √} = √x3 = √3, v} ab" Vab 1 n = b" ·Xab-1 = Vab-i 5 4 Ans. √3a. 4. Reduce 4V to its simplest form. Ans. 133. Ans. VV V&×6= √6. = Ans. 6. 3 Ans. 198. (56.) If we have a radical of the form 4 V/15, and wish to find the root as nearly as possible in whole numbers, it is necessary to reverse the process described in the preceding rule, and place the coefficients under the radical sign. This is done by squaring the coefficient, and multiplying the radical by its square: thus, 4 v15 V/16×15 = √240. Now the entire part of the square root of 210 is 15; but if we had taken the square root of 15, the number under the radical sign, we should have found 3 for the entire part of the root, which, multiplied by 4, would give only 12 for the value of the radical in whole numbers. So also, 6 √/20 = 6 ×4=24 : but by putting the coefficient under the sign, we have, 6 V20 √36 x0 = √/720 = 26. = The same principle may be applied to any radical whatever; and hence, for putting the coefficient of a radical under the sign, we have the following RULE. Raise the coefficient to a power corresponding to the index of the radical, and multiply the radical by this power. 3. (a+b)vab= √(a+b)2 × ab = va3b+2a2b2+ab3. To reduce radicals of different indices, to equivalent radicals, having a common index. RULE. (57.) Find the least common multiple of all the indices; this will be the common index. Divide this common index by each particular index, and raise the quantity under the radical sign, to the power denoted by the quotient. 2 3 4 Let it be required to reduce vac, 3b, and va3b, to equivalent radicals having a common index. The least common multiple of the indices is 12; and if we divide this by each particular index, we obtain, for the quotients, respectively, 6, 4, and 3. The first radical is therefore to be raised to the sixth power, the second to the fourth, and the third to the 12 12 12 third, and they become, Vac, V81b, and ab3. This rule depends upon the principle, that the value of a radical is not altered, when the index of the radical, and the exponent of the quantity under the sign, are both multiplied by the same number. To prove this, we have only to show that the Va" is the same as the Va. We have seen (45) that the root of a quantity may be extracted by dividing the exponent of the quantity by the index of the root. Accord ing to this notation, the mnth root a", is written an, which 1 m is the same as aTM, which again is equivalent to a ; whence 2. Reduce V and V to a common index. 6 8cs a3 4. Reduce va, vb, Ve, and Vd to a common index. 5. Reduce V, V, and √2 to a common index. 3 6. Reduce 3 and 5 to a common index. (58.) If the radicals are similar, add or subtract their coefficients; but if they are not similar, the addition or subtraction can only be indicated. EXAMPLES. 1. Find the sum of a V2ac, 2√2ac, and 3a V2ac. Ans. (4a+2) V2ac. 3 2. Find the sum of 3a 2bc, 7 V11a2, and 126 v19c2. 3 3 Ans. 3a v2bc +7 V11a2 + 12b √19c2. |