INVOLUTION is the raising of powers from any proposed root; or the method of finding the square, cube, biquadrate, &c. of any given quantity. RULE I. Multiply the index of the quantity by the index of the power to which it is to be raised, and the result will be the power required. Or multiply the quantity into itself as many times less one as is denoted by the index of the power, and the last product will be the answer. Note. When the sign of the root is +, all the powers of it will be +; and when the sign is, all the even powers will be +, and the odd powers: as is evident from multiplication (m). (m) Any power of the product of two or more quantities is equal to the same power of each of the factors multiplied together. And any power of a fraction is equal to the same power of the numerator divided by the like power of the denominator. Also, am raised to the nth power is amn; and - a raised to the ath power is amn, according as n is an even or an odd number. EXAMPLES FOR PRACTICE. 1. Required the cube or third power, of 2a2. 2 3. Required the cube, or third power, ofx2y3 3 За2 x 4. Required the biquadrate, or 4th power of 5. Required the 4th power of a+x; and the 5th power of a-y. RULE II. A binomial or residual quantity may also be readily raised to any power whatever, as follows : 1. Find the terms without the coefficients, by observing that the index of the first, or leading quantity, begins with that of the given power, and decreases continually by 1, in every term to the last; and that in the following quantity, the indices of the terms are 1, 2, 3, 4, &c. 2. To find the coefficients, observe that those of the first and last terms are always 1; and that the coefficient of the second term is the index of the power of the first : and for the rest, if the coefficient of any term be multiplied by the index of the leading quantity in it, and the product be divided by the number of terms to that place, it will give the coefficient of the term next following. Note. The whole number of terms will be one more than the index of the given power; and when both terms of the root are +, all the terms of the power will be +; but if the second term be, all the odd terms will be +, and the even terms ; or, which is the same thing, the terms will be + and - alternately (n). (n) The rule here given, which is the same in the case of integral EXAMPLES. 1. Let a+x be involved, or raised to the 5th power. Here the terms, without the coefficients are α5, α4x, α3x2, α2x3, αx4, x5. And the coefficients, according to the rule, will be 1, 5, 5×4 10×3 10+2 5×1 Whence the entire 5th power of a+x is a5+5a4x+10a3x2+10a2x3+5ax4+x5 2. Let a-x be involved, or raised, to the 6th power. Here the terms, without their coefficients, are αβ, αχ, α4x2, 3x3, a2x4, αx5, x. And the coefficients, found as before, are Whence the entire 6th power of a-x is 6 1. ; α6-6a5x+15a4x -20a3x3+15a2x4-6ax5 +x powers as the binominal theorem of Newton, may be expressed in general terms, as follows: (a+b)mam+mam-16+m. (a-b)m=am-mam-1b+m. m-1 2 m-1 2 m-1 m-2 -am-262+m.-2 -am-262-m. 3 m-1 m-2 2 3 am-363, &c am-363, &c. which formulæ will, also, equally hold when mis a fraction, as will be more fully explained hereafter. It may, also, be farther observed, that the sum of the coefficients in every power, is equal to the number 2 raised to that power. Thus 1+1-2, for the first power; 1+2+1=4-22, for the square; 1+3+3+1-8-23, for the cube, or third power; and so on. 3. Required the 4th power of a+x, and the 5th power of a-x. 4. Required the 6th power of a+x, and the 7th power of a-y. 5. Required the 5th power of 2+x, and the cube of a-bx+c. EVOLUTION. EVOLUTION, or the extraction of roots, is the reverse of involution, or the raising powers; being the method of finding the square root, cube root, &c. of any given quantity. CASE I. To find any root of a simple quantity. RULE. Extract the root of the coefficient for the numeral part, and the root of the quantity subjoined to it for the literal part; then these, joined together, will be the root required. And if the quantity proposed be a fraction, its root will be found, by taking the root both of its numerator and denominator. Note. The square root, the fourth root, or any other even root, of an affirmative quantity, may be either + or Thus, a2 + a or-a, and 64= + bor-b, &c. But the cube root, or any other odd root, of a quantity, will have the same sign as the quantity itself. Thus, 3 Va3=a; -a3=-a; and 5/-a5=-a, &c. (0) (0) The reason why + a and - a are each the square root of a2 is obvious, since, by the rule of multiplication, (+a)×(+) and (-a)x(-a) are both equal to a?. |