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la ihe sixth power,
co-efficient of a in the 4th term x its index
number of terms to that place
the co-efficient of the fifth terin.
\Vhere 20 is the co-efficient of a in the 4th term, 3 is its index ; and 4 denotes the number of terms.
(51). Thus are we furnished with a general rule for raising the 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 dowii shews us that
Obs. 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 terin, 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.
Thus, The first five co-efficients are 1, 9, 36, 84, 126; yl') And the last five ......
} 126, 34, 36, 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,
The last ...
n(n-1) (1-2) Or, (a + b)* = a* + nan-16+ am-? +
n(n-1) By the same process, (a−b)"za"-na-+ andfe
2 n(n-1) (n-2)
*a-318 + &c.; the signs of the terms being alter2 3 nately + and
Er. 1. Raise .r? + 3y2 to the fifth power.
On a comparison of (r + 3y) with (a + b)", (Art. 52) we have, a =?', b = 3y, n = 5.
Now, substituting these quantities for a, b, n, in the foregoing general formula, it is evident that
.....is (ra)'.. =r.
The first }..
.. is (3y)'=243y". So that (x' +372)'=r!+15rRy' +90x®y*+270x*y®+405.xoy® +243 y**
(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.
Er. 2. Raise 3.7 to 2y to the sixth power.
(54.) By means of the rule just laid down, we are enabled to raise a trinomial or a quadrinomial quantity to any power, without the process of actual multiplication.
Illus. Suppose it were required to square a + b + c; placing a vinculum over a + T, and viewing it as one quantity, we shall have (a +6+C) =((a+6) + c) (a+b)* + 2. (a+b.) c + c c' +b .
b) a' + 2ab + 2.4 to 2ac + 2lc + d.
Examples. 1. Raise x-a to the 4th power. 2.
a-l to the 7th power. 3. Icy to the gth power. 4. t + a to the 10th power. 5. 1-2y to the 7th power. 6. a + 2 + c to the 2d power. 7. a + b + c to the 3d power. 8. I + y + 3z to the 2d power.
(55.) Evolution, or the extracting of roots, is the reverse of Involution, and is performed by inquiring what quantity, multiplied by itself till the number of factors amounts to the num. her of units in the index of the given root, will generate the given quantity.
Ols. By reviewing the tables of roots and their powers in Involution (Art. 48), we find that the cube root of 216 is 6, because 7 multiplied into itself till the number of factors amounts to three, will give 216.- Also,
5 X 5 X 5 X 5 625, :, the 4th root of 625 is 5.
XOXO - 18 .. the cube root of -39 is -6.
square root of is
26 From these data, we discover that the square, cube, &c. roots of simple and compound quantities, necessarily divide Evolution into several cases.
To extract the roots of simple quantities.
(56.) Rule. Extract the root of the co-efficient for the numerical part, and divide the index of the letter, or letters, by the exponent of ibe proposed root, and it will give the answer required.
Note. Any even ruot of an affirmative quantity may be either + or -, thes the square root of t az is cither + , or - a; for (+a)xita)-ta', and (-a)X(-a) – +a. (Ilins. Art. 29. p. 7.)
And an odd root of any quantity will have the same sign as the quantity itself: thus the cube root of +43 is ta, and the cube root of — az is - a; for (+a)x(+a)x(+a)= +, and (-:)X(-a)X(-a) -a'. (Art. 29.)
Any even root of a negative quantity is impossible; for neither (+a)x(+a fier (-a)X(-a) can produce —u. (Art. 29.)
The nth root of a product is eqnal to the nth root of each of the factors multiplied togсther.
And the nth root of a fraction is equal to the nth root of the numerator divided by the nth root of the denominator.
To find the square root of a compound quantity. (57) Rule.- Range the quantities according to the elimensions of one of the letters, and set the root of the first term in the quotient.
2. Subtract the square of the root, thus found, from the firs term, and bring down the next two terms to the remainder for a dividend.
3. Divide the dividend by double the root, and set the result bord in the quotient and divisor,
4. Multiply the divisor, thus increased, by the term last put in the quotient, and subtract the product from the dividend; and so on, as in common arithmetic.
Examples. | Extract the square rooi of r* - 419 +6ro - 4r+1.
r* -4.79 +6r? -4.x + 1(x2–2x+1=root
21"-2.r) -42° +6r.
2r? -4x+1)2x* --4x+1
2. Extract the square root of 4a* + 12a’r+ 13a*r? + Oars tré.
4a* + 12ax + 13a?r? +- Oax' + x*(2a2 +3artre
4. Required the square root of zo–22° +7.724 + 16
3. Required the square root of a* + 4a’x + ba?r? + 4ar' +7.
Ans, a? + 200 + 4+
Ans. r-r+*. 5. It is required to find the square root of a? +22.
zo Ans. at
&c 2a 8a 16a5
89 6. Required the square root of 4.x* +6x9+ -21% +152 +25
Ans. 2r ++:+5.
To find the roots of powers in general.
(58.) Rule.-1. Find the root of the first term, and place it in the quotient.
2. Subtract its power from that term, and then bring down the second term for a dividend
3. Involve the root, last foun., to the next lowest power, and multiply it by the index of the given power for a divisor.
4. Divide the dividend by this divisor, and the quotient will be the next term of the root