be reduced to an improper frac-* - .** tion. - - & a;-a-1 . . . 7. 1 - —I- be reduced to an improper fraction. 8. Let 1 +*-*.* be reduced to an improper fraction. CASE IV. To reduce an improper fraction to a whole or mixed quantity. RULE. Divide the numerator by the denominator, for the in. part, and place the remainder, if any, over the enominator, for the fractional part ; then the two, joined together, with the proper sign between them, will give the mixed quantity required. EXAMPLES. - 27 - - - - whole quantity. to a ab 3. It is required to reduce the fraction mixed quantity. Involution is the raising of powers from any proposed root ; or the method of finding the square, cube, biquadrate, &c. of any given quantity. 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 ene as is denoted by the index of the power, and the last product will be the answer. JNote. 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 nth power is + amn, according as n is an even or an odd number. ExAMPLES FOR PRACTICE. 1. Required the cube or third power, of 24”. 3. Required the cube, or third power, of-oy" 4. Required the biquadrate, or 4th power of *:::: 5. Required the 4th power of a-Ha ; 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. JNote. 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-Ha be involved, or raised to the 5th power. powers as the binominal theorem of Newton, may be expressed in general terms, as follows: which formulae will, also, equally hold when m is a fraction, as will be more filly 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+2+1=8=23, for the cube, or third power; and so Ott. ~ |