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(45.) To Multiply fractional quantities.
Rule. Multiply the numerators together for a new numerator, and also the denominators for a new denominator ; and reduce the products to their lowest term.
Eramples. 2r 42 1. Multiply by
7 9 Here 28 x 41 = 8.x® the new Num and 7 X 9
.. the fraction required is 8x8
(46.) To divide fractional quantities.
. Invert the divisor, and proceed, in all respects, as in Mul. tiplication. (Arto 45.)
x + b
3 Invert the divisor and it becomes ;
22 14.7 3 42.ro hence Х
(dividing the numerator and 9 2.7 18.r 3 denominator by 6.r) is the fraction required.
x + a 2. Find the quotient of
divided by 23 2b
50 + a + a Here 5.r ta 5х2 + баr + a*
= the quotient 28 - 26
26* required. 141--3 10x 4
20 4. Divide by
1508 3. Divide
15a'b_151,5 6. Divide
12 7. Divide
gr 8. Divide
r (47.) SCHOLIA 1. If the fractions to be divided have a common denominator take the numerator of the dividend for a new numerator, and the numerator
3 of the divisor for the denominator; for it is evident, that are contained
12 9 as many times in
as there are 3's in nine ; i. e. thrice. 2. When a fraction is to be divided by any quantity, it is the same thing whether the numerator be divided by it, or the denominator multiplied by it. For, if it be proposed to divides by c, we change it into , and then di
bc viding the ac by c, we have for the quotient sought. . Hence, when the
bc fraction is to be divided by c, we have only to x the denominator by
5 that quantity, and leave the numerator as it is. Thus * 3 gives
9 - 5 16 80°
3. When the two numerators or the two denominators are divisible by a common quantity, we expunge that quantity from each, and use the quotients in place of the fractions first proposed. 4. The signs + and preserve the same laws in division as in multipli
3 3 cation ; for
(18.) Involution, or the raising of quantities to a given power, is performed by the continued multiplication of a quantity into itself, till the number of factors amounts to the pumber of units in the index of the given power; or it is the method of finding the square, cube, biquadrate, &c. of any given quantity or root.
Ols. These operations may be easily performed upon small numbers and simple algebraic quantities, as will appear fron) the mere inspection of the following tables, which also illustrate the definition we have given of this branch of algebraic computation.
4th Power 116 81 | 256 025 1296 2401 | 4096 | 6561
5th Power 1 32 243 1024312517776 16807 | 32768 | 59049 |100000
Illus.-Jf I want to find the tifth power of the number 6, I proceed thus
216 3d power, or cube.
Nlus. If it is required to fiud the 5th power of - b, ibea
6 the root.
b = + 62 = the square. bx
63 the cube. -bx bx Вх b + 64 * the 4th power. -OX
65 == the 5th power.
2a And the 5th power of the fraction is expanded thus :
Note. When the root is t, all the powers of it will be positive, and when it is negatire the odd powers will take the sign –, and the eren powers the sigo +; and from these tables we derive the following rule for finding the power of any quantity.
(49.) Rule. Multiply, or involve, the quantity into itself to as many factors as there are units in the inder, and the last product will be the power required.
Or, Multiply the index of the quantity by the index of the power, and the result will be the same as before.
Examples. 1. Required the cube of 2a'. 2.
the 4th power of 22°r. 3.
the 3d power of 8rRy. 4.
the 4th power of a + 3b. 5. the 5th power
the square of 37% + 2x + 5. 7.
the square of a + 2z. 8.
the cube of 3.7 5. 9.
the square of x + y + v. 10.
the cube of 6roy + 12ry. 11.
the square of x? ys + %. 12.
the 5th power of as + b 2 + cs.
(60.) SCHOLIUM. In the involution of a binomial quantity of the form o + b the component terms of the successive powers will be found to bear a certain relation to each other, and to observe a certain law, according to the inces of the given power. To illustrate this, we inspect the following table :
Obs. The successive powers of a- t are the same as those of (+b, except that the sign of the terms will be alternately + and
Illus. In examining that column of the foregoing table which is oecupied by the expanded powers of a + I, we discover,
1. That in each case, the first term is a raised to the given power ; and the last term is b raised to the given power. For in the square, od is the first, and be is the last tern : in the cube, a' is the first, and the last, &c. 2. That in the intermediate terms, the powers of a decrease, while
powers of 6 increase, by unity, in each successive term. 3. That in each case, the co-efficient of the second term is the same with the index of the given power.
4. That if the co-efficient of a in any term be x by its index, and the product + by the number of terms to that place, the quotient will give the co-efficient of the next term.
For example, in the 4th power, the
co-efficient of a in the ad term X, its index=4***
6 = co-efficient of the 3d term.
2 Where 4 is the co-efficient of a, 3 is its itider, and 2 denotes the number of terms.