3. Required the 4th power of a-Hz, and the 5th power of a-a.

4. Required the 6th power of a-Hz, and the 7th power of a-y.

5. Required the 5th power of 2+2, and the cube of a—bac-Hc.


Evolution, or the extraction of roots, is the reverse of involution, or the raising powers; being the method of finding the square root, cube root, &c. of any given quantity.


To find any root of a simple quantity,

Extract the root of the coefficient for the numeral part, and the root of the quantity subjoined to it for the literal part; then these, joined together, will be the root required. And if the quantity proposed be a fraction, its root will be found, by taking the root both of its numerator and denominator. JNote. The square root, the fourth root, or any other even root, of an affirmative quantity, may be either + or —. Thus, va” = + a or—a, and 3/b" = + b or—b, &c. But the cube root, or any other odd root, of a quantity, will have the same sign as the quantity itself. Thus, $/a”-a; 3/-a8=—a ; and R/-a” =—a, &c.(0)

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It may here, also, be farther remarked, that any even root of a negative quantity, is unassignable.

Thus, y-a” cannot be determined, as there is no quantity, either positive or negative, (+ or –), that, when multiplied by itself, will produce — a”.

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12 of 2 2. It is required to find the square root of # and

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Aco T A/4c” 26
. It is required to find the square root of 4a2.c".
. It is required to find the cube root of— 125a 92°.
It is required to find the 4th root of 256a 428.

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- 4 It is required to find the square root of 4a". 94.2 y? - - 803 7. It is required to find the cube root of ;...1253:6 - 5 on 1 8. It is required to find the 5th root of — *:::::

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To extract the square root of a compound quantity.


1. Range the terms, of which the quantity is composed, according to the dimensions of some letter in them, beginning with the highest, and set the root of the first term in the quotient.

2. Subtract the square of the root, thus found from the first term, and bring down the two next terms to the remainder for a dividend.

3. Divide the dividend, thus found, by double that part of the root already determined, and set the result both in the quotient and divisor.

4. Multiply the divisor, so increased, by the term of the root last placed in the quotient, and subtract the Product from the dividend; and so on, as in common arithmetic.

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Note. When the quantity to be extracted has no exact root, the operation may be carried on as far as is thought necessary, or till the regularity of the terms shows the law by which the series would be continued.

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Here, if the numerators and denominators of the two last terms be each multiplied by 3, which will not alter their values, the root will become 2: 3:2 33.3 3.5 acA 1-H--- -------- +: -" 3.4", 10 where the law of the series is manifest.

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Find the root of the first term, which place in the quotient ; and having subtracted its corresponding power from that term, bring down the second term for a dividend. * Divide this by twice the part of the root above determined, for the square root ; by three times the square of it, for the cube root, and so on ; and the quotient will be the next term of the root

Involve the whole of the root, thus found, to its proper power, which subtract from the given quantity, and

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