252. Evolution is the opposite of involution. One is finding a power of a quantity, by multiplying it into itself. The other is finding a root, by resolving a quantity into equal factors. A quantity is resolved into any number of equal factors, by dividing its index into as many equal parts; (Art. 241.)
Evolution may be performed, then, by the following general rule;
Divide the index of the quantity by the number expressing the root to be found.
Or, place over the quantity the radical sign belonging to the required root.
1. Thus the cube root of a6 is a2. For a2.a2.a2 = a6.
Here 6, the index of the given quantity, is divided by 3, the number expressing the cube root.
2. The cube root of a or a1, is a1/3 or 3√a.
For a1/3.a1/3.a1/3r or 3√a.3√a.3√a = a. (Arts 239,242).
3. The 5th root of ab, is (ab)1/5 or 5√ab.
4. The n-th root of a2 is a2/n or n√5. The 7th root of 2d - x, is (2d - x)1/7 or 7√ .
6. The cube root of a1/2, is a1/6. (Art. 160.)
7. The 4th root of a-1 is a-1/4.
8. The cube root of a2/3 is a2/9.
9. The n-th root of xm, is xm/n.
253. According to the rule just given, the cube root of the square root is found, by dividing the index 1/2 by 3, as in example 6th. But instead of dividing by 3, we may multiply by 1/3. For (1/2):3 = (1/2):(3/1) = (1/2).(1/3). (Art. 159.)
So (1/m):n = (1/m).(1/n). Therefore the roth root of the n-th root of a is equal to a(1/n).(1/m).
That is, (a1/n)1/m = a(1/n).(1/m) = a1/mn.
Here the two fractional indices are reduced to one by multiplication.
It is sometimes necessary to reverse this process; to resolve an index into two factors.
Thus x1/8 = x(1/4).(1/2) = (x1/4)1/2. That is, the 8th root of x is equal to the square root of the 4th root.
So (a + b)1/mn = (a + b)(1/m).(1/n) = [(a + b )1/m]1/n.
It may be necessary to observe, that resolving the index into factors, is not the same as resolving the quantity into factors. The latter is effected, by dividing the index into parts.
254. The rule in Art. 252, may be applied to every case in evolution. But when the quantity whose root is to be found, is composed of several factors, there will frequently be an advantage in taking the root of each of the factors separately.
This is done upon the principle that the root of the product of severed factors, is equal to the product of their roots.
Thus √ab = √a.√b. For each member of the equation if involved, will give the same power.
The square of √ab is ab. (Art. 237.)
The square of √a.√b, is √a.√a.√b.√b. (Art. 99).
But √a.√a = a. (Art. 237.) And √b.√b = b.
Therefore the square of √a.√b = √a.√a.√b.√b = ab, which is also the √ab.
On the same principle, (ab)1/n =a1/n.b1/n.
When, therefore, a quantity consists of several factors, we may either extract the root of the whole together; or we may find the root of the factors separately, and then multiply them into each other.
Ex. 1. The cube root of xy, is either (xy)1/3 or x1/3.y1/3.
2. The 5th root of 3y, is 5√3y or 5√3.5√y.
3. The 6th root of abh, is (abh)1/6, or a1/6b1/6h1/6.
4. The cube root of 8b, is (8b)1/3, or 2b1/3.
5. The n-th root of xny, is (xny)1/n or xy1/n.
255. The boot of a fraction is equal to the root of the numerator divided by the root of the denominator.
1. Thus the square root of a/b = a1/2/b1/2. For (a1/2/b1/2).(a1/2/b1/2) = a/b.
2. The square root of x/ay, √x/√ay.
3. √ah/√xy= √
256. For determining what sign to prefix to a root, it is important to observe, that
An odd root of any quantity has the same sign as the quantity itself.
An even boot of an affirmative quantity is ambiguous.
An even root of a negative quantity is impossible.
That the 3d, 5th, 7th, or any other odd root of a quantity must have the same sign as the quantity itself, is evident from Art. 215.
257. But an even root of an affirmative quantity may be either affirmative or negative. For, the quantity may be produced from the one, as well as from the other. (Art 215.)
Thus the square root of a2 is +a or -a.
An even root of an affirmative quantity is, therefore, said to be ambiguous, and is marked with both + and -.
Thus the square root of 3b, is ±√3b.
The 4th root of x, is ±x1/4.
The ambiguity does not exist, however, when, from the nature of the case, or a previous multiplication, it is known whether the power has actually been produced from a positive or from a negative quantity.
258. But no even root of a negative quantity can be found. The square root of-a2 is neither -a nor +a.
For +a.+a = +a2.
And -a.-a = +a2 also.
An even root of a negative quantity is, therefore, said to be impossible or imaginary.
There are purposes to be answered, however, by applying the radical sign to negative quantities. The expression √-a is often to be found in algebraic processes. For, although we are unable to assign it a rank, among either positive or negative quantities; yet we know that when multiplied into itself, its product is -a, because √-a is by notation a root of -a, that is, a quantity which multiplied into itself produces -a.
This may, at first view, seem to be an exception to the general rule that the product of two negatives is affirmative. But it is to be considered, that √-a is not itself a negative quantity, but the root of a negative quantity.
The mark of subtraction here, must not be confounded with that which is prefixed to the radical sign. The expression √-a is not equivalent to -√a. The former is a root of -a; but the latter is a root of+a:
For -√a.-√a = √aa = a.
The root of -a, however, may be ambiguous. It may be either +√-a, or -√-a.
One of the uses of imaginary expressions is to indicate an impossible or absurd supposition in the statement of a problem. Suppose it be required to divide the number 14 into two such parts, that their product shall be 60. If one of the parts be x, the other will be 14 - x. And by the supposition,
x.(14 - x) = 60, or 14x - x2 = 60.
This, reduced, by the rules in the following section, will give
x = 7 ±√-11.
As the value of x is here found to contain an imaginary expression, we infer that there is an inconsistency in the statement of the problem: that the number 14 cannot be divided into any two parts whose product shall be 60.
259. The methods of extracting the roots of compound quantities are to be considered in a future section. But there is one class of these, the squares of binomial and residual quantities, which it will be proper to attend to in this place. It has been shown (Art. 210,) that the square of a binomial quantity consists of three terms, two of which are complete powers, and the other is a double product of the roots of these powers. The square of a + b, for instance, is
a2 + 2ab + b2,
two terms of which, a2 and b2 are complete powers and 2ab is twice the product of a into b, that is, the root of a2 into the root of b2.
Whenever, therefore, we meet with a quantity of this description, we may know that its square root is a binomial; and this may be found, by taking the root of the two terms which are complete powers, and connecting them by the sign +. The other term disappears in the root. Thus, to find the square root of
x2 + 2xy + y2,
take the root of x2, and the root of y2 and connect them by the sign +. The binomial root will then be x + y.
In a residual quantity, the double product has the sign - prefixed, instead of +. The square of a - b, for instance, is a2 - 2ab + b2 (Art. 210.) And to obtain the root of a quantity of this description, we have only to take the roots of the two complete powers, and connect them by the sign -. Thus the square root of x2 -2xy + y2 is x - y. Hence,
260. To extract a binomial or residual square root, take the roots of the two terms which are complete powers, and connect them by the sign which is prefixed to the other term.
Ex. 1. To find the root of x2 + 2x + 1.
The two terms which are complete powers are x2 and 1.
The roots are x and 1. (Art. 244.)
The binomial root is, therefore, x + 1.
2. The square root of a2 + a + 1/4, is a + 1/2. (Art. 220.)
3. The square root of a2 + ab + b2/4, is a + b/2.
4. The square root of a2 + 2ab/c + b2/c2, a + b/c.
261. A root whose value cannot be exactly expressed in numbers, is called a surd.
Thus √2 is a surd, because the square root of 2 cannot be expressed in numbers, with perfect exactness.
In decimals, it is 1.41421356 nearly.
But though we are unable to assign the value of such a quantity when taken alone, yet by multiplying it into itself, or by combining it with other quantities, we may produce expressions whose value can be determined. There is, therefore, a system of rules generally appropriated to surds. But as all quantities whatever, when under the same radical sign, or having the same index, may be treated in nearly the same manner; it will be most convenient to consider them together, under the general name of Radical Quantities; understanding by this term, every quantity which is found under a radical sign, or which has a fractional index.
262. Every quantity which is not a surd, is said to be rational. But for the purpose of distinguishing between radicals and other quantities, the term rational will be applied, in this section, to those only which do not appear under a radical sign, and which have not a fractional index.