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To multiply fractional quantities together
Multiply the numerators together for a new numerator, and the denominators for a new denominator ; and the former of these, being placed over the latter, will give the product of the fractions, as required (k).
1. It is required to find the product of an
the product required.
XX 4x X 10x 40x3
atx 3. It is required to find the product of and
(k) When the numerator of one of the fractions to be multipli. ed, and the denominator of the other, can be divided by some quantity which is common to each of them, the quotients may be used instead of the fractions themselves,
Also, when a fraction is to be multiplied by an integer, it is the same thing whether the numerator be multiplied by it, or the denominator divided by it. Or if an integer is to be multiplied by a fraction, or a fraction by an integer, the integer may be considered as having unity for its denominator, and the two be then multiplied together as usual.
3x 5x 4. It is required to find the product of and
2x 6. It is required to find the continued product of
atx 7. It is required to find the continued product of 2x 3ab Бас
9. It is required to find the continued product
-1 and 2a
10. It is required to find the continued product of a2 x2 a3 - bi
and at atbax+x2
To divide one fractional quantity by another.
Multiply the denominator of the divisor by the numerator of the dividend, for the numerator ; and the nume. rator of the divizor by the denominator of the dividend, for the denominator. Or, which is more convenient in practice, multiply the dividend by the reciprocal of the divisor, and the product will be the quotient required. (1)
2x 1. It is required to divide by
9 9x 3
sta xt6 3. It is required to divide
b 5x ta
a3 +23 sta
Х -a3 +33 x (a3+x3) 2 --ax+a?
73 3 5. It is required to divide
(1) When a fraction is to be divided by an integer, it is the same thing whether the numerator be divided by it, or the denominator multip'ied by it.
Also, when the two numerators, or the two denominators, can be divided by some common quantity, that quantity may be thrown out of each, and the quotients used instead of the fractions first proposed.
3+1 2.x 7. It is required to divide by
8. It is required to divide
2ax + x2
X4 - 64 10. It is required to divide
32 - 2x +62
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 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 t, all the powers of it will be t; and when the sign is all the even pow. ers will be t, 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 ; und - a raised to the nth power is amn, according as n is an even or an odd number.