Evolution and Radical Quantities
Art. 236. If a quantity is multiplied into itself, the product is a power. On the contrary, if a quantity is resolved into any number of equal factors, each of these is a root of that quantity.
Thus b is the root of bbb; because bbb may be resolved into the three equal factors, b, and b, and b.
In subtraction, a quantity is resolved into two parts.
In division, a quantity is resolved into two factors.
In evolution, a quantity is resolved into equal factors.
237. A root of a quantity, then, is a factor, which multiplied into itself a certain number of times , will produce that quantity.
The number of times the root must be taken as a factor, to produce the given quantity, is denoted by the name of the root.
Thus 2 is the 4th root of 16; because 188.8.131.52 = 16, where two is taken four times as a factor, to produce 16.
So a3 is the square root of a6; for a3.a3 = a6. (Art. 229.)
And a2 is the cube root of a6; for a2.a2.a2 = a6.
And a is the 6th root of a6; for a.a.a.a.a.a = a6.
Powers and roots are correlative terms. If one quantity is a power of another, the latter is a root of the former. As b3 is the cube of b, b is the cube root of b3.
238. There are two methods in use, for expressing the roots of quantities; one by means of the radical sign √, and the other by a fractional index. The latter is generally to be preferred; but the former has its uses on particular occasions.
When a root is expressed by the radical sign, the sign is placed over the given quantity, in this manner, √a.
Thus 2√a is the 2d or square root of a.
3√a is the 3d or cube root.
n√a is the n-th root.
And n√a + y si the n-th root of a + y.
239. The figure placed over the radical sign, denotes the number of factors into which the given quantity is resolved; in other words, the number of times the root must be taken as a factor to produce the given quantity.So that √a.√a = a.
And 3√a.3√a.3√a = a.
And n√a.n√a.n√a.... n times = a.
The figure for the square root is commonly omitted; √a being put for 2√a. Whenever, therefore, the radical sign is used without a figure, the square root is to be understood.
240. When a figure or letter is prefixed to the radical sign, without any character between them, the two quantities are to be considered as multiplied together.
Thus 2√a, is 2.√a that is, 2 multiplied into the root of a, or, which is the same thing, twice the root of a.
And x√b, is x.√b, or x times the root of b.
When no coefficient is prefixed to the radical sign, 1 is always to be understood; √a being the same as 1.√a, that is, once the root of a.
241. The method of expressing roots by radical signs, has no very apparent connection with the other parts of the scheme of algebraic notation. But the plan of indicating them by fractional indices is derived directly from the mode of expressing powers by integral indices. To explain this, let a6 be a given quantity. If the index be divided into any number of equal parts, each of these will be the index of a root of a6.
Thus the square root of a6 is a3. For, according to the definition, (Art. 237,) the square root of a6 is a factor, which multiplied into itself will produce a6. But a3.a3 = a6. (Art. 229.) Therefore, a3 is the square root of a6. The index of the given quantity a6, is here divided into the two equal parts, 3 and 3. Of course, the quantity itself is resolved into the two equal factors, a3 and a3.
The cube root of a6 is a2. For a2.a2.a2 = a6.
Here the index is divided into three equal parts, and the quantity itself resolved into three equal factors.
The square root of a2 is a1 or a. For a.a = a2.
By extending the same plan of notation, fractional indices are obtained.
Thus, in taking the square root of a1 or a, the index 1 is divided into two equal parts, 1/2 and 1/2 ; and the root is a1/2.
On the same principle,
The cube root of a, is a1/3 = 3√a.
The n-th root, is a1/n=n√a.
And the n-th root of a + x, is (a + x)1/n = n√a.
242. In all these cases, the denominator of the fractional index, expresses the number of factors into which the given quantity is resolved.
So that a1/3.a1/3.a1/3 = a. And a1/n.a1/n.a1/n..... n times = a.
243. It follows from this plan of notation, that
a1/2.a1/2 = a1/2+1/2. For a1/2+1/2 = a1 or a.
a1/3.a1/3.a1/3 = a1/3+1/3+1/3 = a1.
where the multiplication is performed in the same manner as the multiplication of powers, (Art. 230,) that is, by adding the indices.
244. Every root as well as every power of 1 is 1. (Art. 205.) For a root is a factor, which multiplied into itself will produce the given quantity. But no factor except 1 can produce 1, by being multiplied into itself.
So that 1n, 1, √1, n√1. are all equal.
245. Negative indices are used in the notation of roots, as well as of powers. See Art. 203.
Thus1/a1/2 = a-1/2; 1/a1/3 = a-1/3; 1/a1/n = a-1/n.