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Involution

root, quantity, power, raised, negative and am

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INVOLUTION. If a quantity be conti nually 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, aXa, or a', is called the setond power of a; axaXa, or 0, the third pow. er, xa....(a), or an, the nth power.

If the quantity to be involved be nega. tive, the signs of the even powers will be positive, and the sig-ns of the odd power negative.

For-a x-a=as;-ax-aX - a - &c.

A simple quantity- is raised to any pow er, by multiplying the index of every fac tor in the quantity- by the exponent of the power, and prefixing the proper sign de termined by the last article.

'rhus, on raised to the nui power is am", Because .x1 v am x am ....to n factors, by the rule of multiplication, is amn ; also, 47Pn=-a bxa bx&e. to n factors, or a a....to n factors xbXb X b-to n factors =_-hnxbn ; and a' b3 e raised to the fifth power is am 105 c5. Also, raised to the nth rrjwer is* am n ; where the positive or negative sign is to be pre fixed; according as n is an even or odd number.

If the quantity to be involved be a frac tion, both the numerator and denomina tor must be raised to the proposed power. If the quantity proposed b e a compound one, the involution may either be repre sented by the proper index, or it may ac tually take place.

Let a+b be the quantity to be raised to any power.

b a+ b b f -a b-l-b2 a >7-bi2 or a2+2 a b+b2 the sq.or 2d power a -Fb a3+2 n2 b+a 62 + n2 b--1-2 a b2+ 63 or a3+3 a2 6+3 a 62+63 the 3d pr. a -1-b a4-1-3 a3 b±3a22-1 b3 a3 6+3 a2 b2-1-3 a D3-Fb4 or a4+4 a3 b-F6 a2.52--F4 a b34-b4 the fourth power.

If b be negative, or the quantity to be involved be a -b, wherever an odd pow er of b enters, the sign of the term must be negative.

Hence, as----44.-_-a4 - 4 a' b ± 6 a' b2 -4 a b3+64.

Evolx-rros or the extraction of roots, is the method of determining a quantity, which, raised to a proposed power, mill produce a given quantity.

Since the nth power of am is amn, the root of ems must be am ; i. r. to ex

tract any root of a single quantity, we must divide the index of that quantity by the index of the root required.

When the index of the quantity is not exactly divisible by the number which ex presses the root to be extracted, that root must be represented according to the no tation already pointed out Thus the square, cube, fourth, root of a' arc respectively represented by (a' + x91, (a' + (a' -I- 1 (a' -I- the same roots of 1 ,are represe nt e dby 31 (a"-i-x") of n.

If the root to be extracted be express ed by an odd number, the sign of th e root will be the same with the sign of the pro posed quantity.

If the root to be extracted be expressed by an even number, and the quantity pro posed be positive, the root may be either positive or negative. Because either a positive or negative quantity, raised to such a power, is positive.

If the root proposed to be extracted be expressed by an even number, and the sign of the proposed quantity be negative, the root cannot be extracted; because no quantity, raised to an even power, can produce a negative result. Such roots are galled impossible.

Any root of a product may be found by taking that root of each factor, and mul tiplying the roots, so taken, together.

2. I Thus, (a b)n = an x b because each of these quantities, raised to the pow er, is a 6.

I ? 2 In a=b, then an x an = and in the r * rfa n n same manner a X a =a .

Any root of a fraction may be found by taking that root both of the numerator and denominator. Thus, the cube root off, is L or and Cr to 2 =— Or 4 b n Xb--n.

To extract the square root of a compound pantie!".

a'+2 a (cr-}-1.

a' 2 a4-b)2 a 2 a 64-6' • • Sizice the square root of a'+2 a 6+62 is a±b, whatever be the values of a and b, we may obtain a general rule for the extraction of the square root, by observ ing in what manner a and b maybe deriv ed from (0+2 a 6+6'.

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