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(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 number 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.
Obs. 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.
Roots and Pouers of Numbers.
5th Power 1 32 243 1024312517776 10807 | 32768 | 59049 |100000 Illus.- If I want to find the tifth power of the number 6, I procecd thus
Illus. If it is required to find the 5th power of - b, ibea
-b the root.
- b + b2 = the squure. ox b x
- 63 = the cube. bx bx bx
+ bx x the 4th power. 6X bx 6 x
65 as the 5th power.
2a And the 5th power of the fraction is expanded thus :
Note. When the root is +, all the powers of it will be positive, and when it is negative the odd powers will take the sign, and the even powers the sigo t; 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 brany 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 2a*r. 3.
the 3d power of - 8rRy'. 4.
the 4th power of a + 3b. .
the 5th power of a - b. 6.
of 3x + 2x + 5. 7.
square of a + 2x. 8.
the cube of 3.3 5. 9.
the square of x + y + v. 10.
the cube of 6x'y + 12xy. 11.
the square of 22 y2 + z. 12.
the 5th power of a + b + c.
(50.) SCHOLIUM. In the involution of a binomial quantity of the form « + be 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 :
5th Power. 27 bis las + 50*6 + 10a%b? + 1022%3 + 5ab6 + 63
6th Power. a + MO 29 +686 +150%j? + 20a%b3 + 150?1* + ab + 20
Obs. The successive powers of a -b are the same as those of +b, except that the sign of the terms will be alternately + and
Nlus. In examining that column of the foregoing table which is occupied by the expanded powers of a + l, 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, a> is the first, and be is the last terni : in the cube, a' is the first, and ps the last, &c.
2. That in the intermediate terms, the powers of a decrease, while tho powers of b 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
number of turns to that place
*6 = co-efficient of the 3d term. Where 4 is the co-efficient of n, 3 is its inder, and 2 denotes the number of terms.
Where 20 is the co-efficient of a in the 4th term, 3 is its inder ; and 4 denotes the number of terms.
(51). Thus are we furnished with a general rule for raising tne binoinal a + b to any power, without the process of actual multiplication. For were we required to raise a + b to the eighth power, the rule just laid down shews us that
Ols. When the number of terms is even in the resulting quantity, the co-efficients of the two middle terms are the same ; and, in all cases, the co-efficients increase as far as the middle term, and then decrease precisely in the same manner, till we arrive at the last term. Guided by this law of the co-efficients, we need only calculate them as far as the middle term, and then set down the remaining ones in an inverted order.
The first five co-efficients are 1, 9, 36, 84, 126; yl') And the last five .....
126, 84, 36, 9, 1. 152.) But this rule may be exhibited in its most general form by the Newtonian Binomial Theorem.
Suppose we were required to raise the binomial a + b to any power denoted by n : From the principles already laid down,
n(n-1) OT, (a + b)* = 2" + na"-" + a-22? +
anmoge + &c. ... 2.3.4
By the same process, (2-6)*=2"-na*-10+
2 n(n-1) (n--2)
23-018 + &c.; the signs of the terms being alter2.3 nately t and
The first }..
Er. l. Raise r? + 3y to the fifth power.
On a comparison of (r' + 3y)' with (a + b)", (Art. 52) we have, Q =, b = 3y?, n = 5.
Now, substituting these quantities for a, b, n, in the foregoing general formula, it is evident that
.is (re)'.. =r term 2d... (non-10)
is 5 x(72)* x 3yo = 153oy®.
ха 2.3.4 Last.... (bn)....
...is (3y')'=243y'o. So that (r? +3y2)'=x10 + 15zRy' +903@y*+270x*y®+4053*y* +243y!"
(53.). We may observe, in the application of this formula, that the number of terms of which the binomial consists, is always one more than the index of the given power; therefore, after having calculated as many terms as there are units in the index of the given power, we may instantly proceed to the last term.