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INVOLUTION. 54. INVOLUTION is the raising of any given quantity to any proposed power ; such as the square, cube, &c.
If a quantity be continually multiplied by itself, it is said to be involved, or raised; and the power to which it is raised is expressed by the number of times the quantity has been employed in the multiplication ; thus a Xa, or a', is called the second power of a; axaxa, or as, the cube, or third power; a Xa....(n), or a", the
RULE. When the quantity or root has no index, its power will be represented by placing the index of the required power above it; thus the fourth power of x is 24; the fourth power of xty is (x+y)“. If the quantity proposed be a compound one, the involution may either be represented by the proper index, or it may actually take place. If the quantity to be involved be negative, the signs of the even powers will be positive, and the signs of the odd powers negative ; for --ax-a=a?; -ax-ax-a=.
If the quantity be a fraction, raise both the numerator and denominator to the same power. When the quantity to be raised is itself a power, multiply the index of the quantity by the index of the power; thus the cube of a® is a:
a3x8 : or generally let m or n represent any powers whatever ; then 2", raised to the nth power is wmx", or "; and —x", raised to any power, n, will give t, plus, or minus, according as n is an odd or even number. If n be used for an uneven number, the sign will be +, if odd —; then —20° to the nth power, will be represented by +=+xm**.
Roots and Powers of Numbers.
The operation is performed in the same manner for simple algebraic quantities, except that in this case it must be observed, that the powers of negative quantities are alternately + and ; the even powers being positive, and the odd powers' negative. Thus the square of +2a is +2aX+2a, or 4a'; the square of
xí xx=>, 4th power =
-2a is.—2aX—2a, or +4a? ; but the cube of —20=2ax-
5 X 3 Б 34
631 5 4y
Cube Square Root
4th power power
243x10 32a5 a10 24325 2015
3125 1024y What is the square of a +26? What is the cube of a-x?
α- a2x +2ab+462
a'x+? Sq. = 4+4ab+462
a Q+4ab+ 4ab?
ao—2a*x+ a za
- a'x+2ax?- _23
well known that the value of the figures in the com-
.. by 100a+10b+c.
55. EVOLUTION is the reverse of Involution, and consists in finding the square, cube, &c. roots of any given quantity.
CASE I. To extract the roots of a simple quantity, or powers.
RULE. Extract the root of the numerical coefficient, if it have any, as in common arithmetic; then divide the index of the given power by 2, for the square root, 3 for the cube root, 4 for the biquadrate root, and depending on the index of the root required;
2 thus the square root 9x'=3x2=3x=3x; and the cube root of
—3x'= 8x6 =2x2x. If the coefficient be a fraction, extract the root both of its nu
4 2 1 1 merator and denominator ; thus, the square = x, or
9 3 2 4 1. Find the square root of 16aʼz=16Xa xam, or
16aor=N 16X Na'xNx=+4Xaxx=+4ax, Ans. 2. Find the cube root of 8aRx•, or (8XQX 203). .: 8a®x=8XaXX32Xaxx=2ax, Ans. 3. Find the cube root of —125ax", or --125 XaXx®. ..-125a®z=-125X Yaxus 5XaXã=5ax 32x1°y.
Ans. 4. Find the 5th root of
4a" 5. Find the square root of 2a*b*, or or 4az9, or
9.xy 6. Find the cube root of -64a88®, or -125a8z®, or
12526 7. Find the fourth root of 81a*b*, or 256a*xx. 3abnb, or 4ax.
32a5710 8. Find the fifth root of -322588, or
8a2b2 4aob? 2 72a2b5N (a?84x2)=ab 2, or N G
303 2ab 2
4a Wä or 4a x®
N4da26 N4Na 2az?
Ans. VINXNÝ Зry
-64a0-4ab*, or -125a®z=125a\r's 5axe or
Ans. $81ab=w (81a*8* X 62) 125x6 125x3x 3abb, or 256a*z* =\/256/d4xVx=4ax®, Ans. Ņ_3286
-32a5210 = (-32a585xb)=2abb, Ans. or $
243 V–325d xx_-2ax®
3 Case II. To extract the square root of a compound quantity. Rule. Arrange the terms according to the power of some letter, as in division. Find the a’+2ab +62+2ac+2bc+(a+b+c root of the first term, and set a? it in the quotient. Subtract 2a+b)2ab+* .
+ the square of the root thus
2ab+ь? found from the first term, and
2a there will be no remainder. 24+26+c) Rac+26c+o Then bring down the next
2ac2bc+o two terms for a dividend, and take double (or two times) the root already found for a divisor. Divide the dividend by the divisor, and put the result both in the quotient and divisor. Then proceed as in common arithmetic.
6x-4x+2|12x4_8x+4 First, we extract the square root of 9x*, the first term. This gives 3x2 for the first term of the root required. This we place on the right hand of the dividend, in the manner of the quotient, as in division. Squaring this term, and subtracting it from the dividend, we get for a remainder -1228+16x2–8x+4 : we now double 3x”, and place it as a divisor on the left of this remainder, and dividing by it, 6x", the first term of the remainder, we get or obtain the quotient -24, the second term of the root sought, which we annex, with the proper sign, to the double root 6x?; multiply the whole of this quantity, 6x2–2x, by 2x, and subtracting the product from the first remainder, we obtain for a second remainder 12x48x+4; then by doubling 3x-2x, the two terms of the root thus found, and dividing 122*, the first term of the new remainder, by 6x*, the first term of the double root, we obtain +2 for a quotient, which is the third term of the root sought; and by annexing it to the double root 6x?_4x, and then multiplying the whole of this quotient, 6x-4x+2, by 2, and subtracting the product from the second remainder, we find 0 for a new remainder, which shows that the root required is 3x2–2x+2.
2. 4x® +12+5x2—24+722–2:+1(2x +32<+1 Mult. 4x
2 4x +32° | 1225-+-5% + *
4x constant trial divis. Mult. 12x +9x4 4x+6.c?—2-414_222 +7.2*
- 4x4–6x8+ 4x +6x-2x+1|4x +6x2–2x+1
Root. 3. 920_1226 +10x4—102+52_2x+1(3x8–2x+x_1 92
2, 6x8_2x")—1228 +100
Constant 6x trial divisor. 1st divis. –12.26 + 4x4
2d divisor. 6.x*— 4x + 22
5. What is the square root of
1+(1+22—42+752 px*+ &c.
x / 8 tot
12 tot * -642 tatrza