That is, the nlh power of a polynomial, is equal to the n"1 power of the first term, plus n times the first term raised to the power ,n—1, multiplied by each of the remaining terms; + other terms of the development. Hence, we see, that the rule for the cube root will become the rule for the n!i root, by first extracting the n'h root of the first term, taking for a divisor n times this root raised to the n— 1 power, and raising the partial roots to the n'h power, instead of to the cube. 2. Extract the 4th root of 16a4—96a2x+216a V—2Wax2 + 8 la4 {2a—3x)4=16a4—96a2x+216aV—216aaJ+81p4l32a2==4x(2a)2 We first extract the 4th root of 16a4, which is 2a. We then raise 2a to the third power, and multiply by 4, the index of the root: this gives the divisor 32a2. This divisor is contained in the second term —96a2x, — Sx times, which is the second term of the root. Raising the whole root to the 4th power, we find the power equal to the given polynomial. 3. Find the cube root of .r8+6x5—40ar'+96a;—64. 4. Find the cube root of 15x*—6x+x"—6x*—20a.3+15x*+1. 5. Find the 5th root of 32s5—80a;4+80a?—40a;2+1 Ox— 1. Calculus of Radicals. 224. When it is required to jextract a certain root of a monomial or polynomial which is not.a perfect power, it can only be indicated by writing the proposed quantity after the sign V7~and placing over this sign the number which denotes the degree of the root to be extracted. This number is called the index of the root, or of the radical. A radical expression may be reduced to its simplest terms, by observing that, the nth root of a product is equal to the product of tie n'h roots of its different factors. Or, in algebraic terms: Vabcd= yaxybxycx For, raising both members to the n'h power, we have for the first, ( V abed) "—abed and for the second, (Vox VbXVcXVd-••)"=( V.a)"-(v/6)"-(Vc)".(v/d)"---=flW Therefore, since the n'h powers of these quantities are equal, the quantities themselves must be equal. Let us take the expression \/5'la4bV, which cannot be replaced by a rational monomial, since 54 is not a perfect cube, and the exponents of a and c are not divisible by 3: but we can put it under the form V54aW= VziaW. V2aV=3ab Vzal?. V8o5=2Va7; V4£aWcs=2ab2cV3acT~; In the expressions, 3ab V-2ac2, 2 VoJ, 2ab2c V3ac3, the quantities placed before the radical, are called co-efficients of the radical. 225. The rule of (Art. 214) gives rise to another kind of simplification. Take, for example, the radical expression, V 4a3; from this rule we have, V4a2— V4a2, and as the quantity affected with the radical of the second degree V, is a perfect square, its root can be extracted, hence Vla2^ Via. In like manner, In general, mVa"="Vyan— Va; that is, when the index of a radical is a multiple of any number n, and the quantity under the the radical sign is an exact power, we can, without changing the value of tJie radical, divide its index by n, and extract the nth root of the quantity under tlie sign. This proposition is the inverse of another, not less important, viz. tee can multiply the index of a radical by any number, provided we raise the quantity under tlve sign to a power of tvhich this number denotes the degree. Thus, Va="Va". For, a is the same thing as Va"; hence, m / tn / "- mil /—— Va— yyo"= Va". This last principle serves to reduce two or more radicals to the same index. For example, let it be required to reduce the two radicals 3V2a and V(a+b) to the same index. By multiplying the index of the first by the index of the second, and raising the quantity 2a to the fourth power; then multiplying the index of the second by' 3, the index of the first, and cubing a-\-b, the values of the radicals will not be changed, and the expressions will become V^=tV~¥aT^,V~l6?-, 4V(a+b) = 'V(a+6)3. 226. Hence to reduce radicals to a common index, we have the following RULE. Multiply the index of each radical by the product of the indices of all the other radicals, and raise the quantity under each radical sign to a power denoted by this product. This rule, which is analogous to that given for the reduction of fractions to a common denominator, is susceptible of some modifications. For example, reduce the radicals Wa, V56, Va2+b*, to the same index. , As the numbers 4, 6, 8, have common factors, and 24 is the most simple multiple of the three numbers, it is only necessary to multiply the first by 6, the second by 4, and the third by 3, and to raise the quantities under each radical sign to the 6th, 4th, and 3d powers respectively, which gives VT= Vrf"; Wb=WF, Va2+6» =V(a2+6?)3.' In applying the above rules to numerical examples, beginners very often make mistakes similar to the following, viz. : In reducing the radicals 1/2 and T/3 to a common index, after having multiplied the index of the first (3), by that of the second (2), and the index of the second by that of the first, then, instead of multiplying the exponent of the quantity under the first sign by 2, and the exponent of that under the second by 3, they often multiply the quantity under the first sign by 2, and the quantity under the second by & Thus, they would have V2"=V2X2=V4; and V^= V3><3= VI Whereas, they should have, by the foregoing rule, VT=V(2)3= VT, and Vr3=V73)r=V27. Reduce V2, V4, W\, to the same index. Addition and Subtraction of Radicals. 227. Two radicals are similar, when they have the same index, and the same quantity, under the sign. Thus, 3 i/ab and 7 Vab, are similar radicals, as also 3a2 and 9c3 3</b3. Therefore, to add or subtract similar radicals, add or subtract their co-efficients, and prefix the sum or difference to the common radical. Thus, 3V6+2V6=5V6, S3Vb-'2Vb=3Vb, 3aV6±2cV6=(3a±2c)V6. =(2a-b) Vb+2a; 3 V4a»+2 V2a=3 V2a+2 V2^=5 V2a. When the radicals are dissimilar, and irreducible, they can only be added or subtracted by means of the signs + or —. . Multiplication and Division. 228. We will first suppose that the radicals have a common index. Let it be required to multiply or divide 'Va by Wb. We have Vax Vb= Vab, and Va Vb=\/ For by raising Va . Wb and Vab to the n'* power, we obtain the same result ab; hence the two expressions are equal. ya "/a a In like manner, —^ and - raised to the n'h power give -j- hence these two expressions are equal. Therefore we have the following RULE. Multiply or divide the quantities under the sign by each other, and give to the product, or quotient, the common radical sign. If they have co-efficients, first multiply or divide them separately. Thus, c d cd or, reducing to its simplest terms, 6a'(a'+b2) 3Vcd |