or third power of 2ax is 8a38x3 ; the fourth power of 4 × 44 - x x x a+b+c is 16 x aa-xx) × a + b + c} *, and the square of the radical quantity axa +x is axa + a. 4. A fraction is involved by raising both the numerator and the denominator to the power proposed. 5. Quantities compounded of several terms are involved by a continual multiplication of all their parts. Thus a + bx a + b = a2 + 2ab † b2. EXAMPLES. (1) Involve or raise x to the fourth power. 5ab (4) Involve to the sixth power. 2c (5) Involve or raise a+b to the sixth power. This is called a binomial root. (6) Involve or raise a-b to the sixth power. (7) Involve or raise a-b to the sixth power. This is called a residual root. There is a rule or theorem, given by Sir Isaac Newton, whereby any power of a binomial, or x-y, may be expressed in simple terms, without the trouble of those tedious multiplications which are required otherwise. Note, m is the exponent of the power, that is, m➡7 in the seventh power, 6 in the sixth power, &c. So that if x-y is to be raised to any power m, the terms, without their coefficients, will be y,x, yx, &c. continued till the exponent of y becomes equal to m: and the coefficients of the respective terms will be So by this theorem any quantity, consisting of two terms, is raised to any power m, with great ease and perspicuity, and will be of great service to the young algebraist, if properly demonstrated to him by his tutor. EVOLUTION, or the extraction of roots, being directly the contrary to involution, or raising of powers, is performed by converse operations, viz. by the division of indices, as involution was by their multiplication. Thus, the square root of x is r3, the cube root of r is r2, also the biquadratic root of x+y) will be x+y): and the cube root of xx-yy will be xx yy. Moreover the square root of xx-y will be xx--yy, its cube root xx-yys, and its biquadratic root xx+yylt, and so of others. Evolution of compound quantities is performed by the following RULE. First, place the several terms, whereof the given quantity is composed, in order, according to the dimensions of some letter therein, as shall be judged most commodious: then let the root of the first term be found, and placed in the quotient; which term being subtracted, let the first term of the remainder be brought down, and divided by twice the first term of the quotient, or by three times its square, or four times its cube, &c. according as the root to be extracted is a square, cubic, or biquadratic one, &c. and let the quantity thence arising be also written down upon the quotient; and the whole be raised to second, third, or fourth, &c. power, according to the aforesaid cases respectively, and subtracted from the given quantity; and if any thing remain, let the operation be repeated, by always dividing the first term of the remainder by the same divisor, found as above. EXAMPLES. (1) It is required to extract the square root of x2+2xy+y2. (2) It is required to extract the square root of x2-2xy+y2. (3) It is required to extract the square root of x1—2x3y+ 3x2y2 — 2xy3+y1. (4) Extract the cube root of x2-6x2y+12xy2+8y3. (5) Extract the biquadratic root of 16x2-96x3y+216x2 y2216xy3+81y1. LXXXVIII. INVOLUTION of SURD QUANTITIES. 1. When the surds are not joined to rational quantities, they are involved to the same height as their index denotes by taking away their radical sign. Thus xx will be x2, and xx+yy will be x2+y2, &c. 2. When surds are joined to rational quantities, involve the rational quantities to the same height as the index of the surd denotes; then multiply the involved quantities into the surd quantities, after the radical sign is taken away, as before. Thus x/yy will be xy, and 4x/xx+yy, will become 16 +16xy; likewise, 2x3/x+y will become 8x+8x°y3, &c. LXXXIX. EQUATIONS. AN equation is when two equal quantities, differently expressed, are compared together, by means of the sign = placed between them. REDUCTION of SINGLE EQUATIONS. RULES. 1. Any term of an equation may be transposed to the contrary side, if its sign be changed. Thus, x+12=20, then will x=20—12=8. 2. If there be any quantity by which all the terms of an equation are multiplied, let them all be divided by that quantity: but if all of them be divided by any quantity, let the common divisor be cast away. b Thus, arab, then will x; also, ifa, by the lat a b ter part of the rule. 3. If there be irreducible fractions, let the whole equation be multiplied by the product of all their denominators; or, which is the same, let the numerator of every term in the equation be multiplied by all the denominators except its own, supposing such terms (if any there be) that stand without a denominator to have a unit subscribed. Thus x++ =11, reduced is 6x+3x+2x=66, or x=6 2 3 2x 4.x per rule 5. Again,+12=+6, this reduced will become 10x+180=12x+90: then per rule 1, x=45. 4. If in your equation there be an irreducible surd, wherein the unknown quantity enters, let all the other terms be transposed to the contrary side (by rule 1.), and then if both sides be involved to the power denominated by the surds, an equation will arise free from radical quantities, unless there happen to be more surds than one'; in which case the operation is to be repeated. Thus x+4=12, by transposition, becomes √x=124=8; which, by squaring both sides, gives x= =64. So likewise, aa+xx-c=b, becomes au+xx=b+c; squared, gives aa-xx=bb+2cb+cc+, then per rule 1. x2=a2+b2+2cd+c2, and x=/¿2+b+2cd+c2. 5. Having by the preceding rules, if there be occasion, cleared your equation of fractional and radical quantities, and so ordered it, by transposition, that all the terms wherein the known quantities are found, may stand on the same side thereof, let the whole be divided by the coefficients, or the sum of the coefficients, of the highest power of the said unknown quantity. 24 Thus, if 6x=24, then will x===4; and if 4x=48—2x, 48 then will 6x=48, per rule 1, and x===8. EXAMPLES, 6 For the learner's exercise in the foregoing rules, set down promiscuously. (1) If 20-3x-8=60-7x, what is the value of x? (2) When 5x-16=3x+12, what is r? 3x 5.x (3) If+5=+2, what is x equal to ? 4 6 |