Powers of Roots
246. It has been shown in what manner any power or root may be expressed by means of an index. The index of a power is a whole number. That of a root is a fraction whose numerator is 1. There is also another class of quantities which may be considered, either as powers of roots, or roots of powers.
Suppose a1/2 is multiplied into itself, so as to be repeated three times as a factor.
The product a1/2+1/2+1/2 or a3/2 (Art. 243,) is evidently the cube of a1/2, that is, the cube of the square root of a. This fractional index denotes, therefore, a power of a root. The denominator expresses the root, and the numerator the power. The denominator shows into how many equal factors or roots the given quantity is resolved; and the numerator shows how many of these roots are to be multiplied together.
Thus a4/3 is the 4th power of the cube root of a.
The denominator shows that a is resolved into the three factors or roots a1/3, and a1/3, and a1/3. And the numerator shows that four of these are to be multiplied together; which will produce the fourth power of a1/3; that is,
a1/3.a1/3.a1/3.a1/3 = a4/3.
247. As a3/2 is a power of a root, so it is a root of a power. Let a be raised to the third power a3. The square root of this is a3/2. For the root of a3 is a quantity which multiplied into itself will produce a3.
But according to Art. 243, a3/2 = a1/2.a1/2.a1/2; and this multiplied into itself, (Art. 100,) is
a1/2.a1/2.a1/2.a1/2.a1/2.a1/2 = a3.
Therefore a3/2 is the square root of the cube of a.
In the same manner, it may be shown that am/n is the mth power of the n-th root of a; or the n-th root of the mth power: that is, a root of a power is equal to the same power of the same root. For instance, the fourth power of the cube root of a, is the same as the cube root of the fourth power of a.
248. Roots, as well as powers, of the same letter, may be multiplied by adding their exponents. (Art 243.) It will be easy to see, that the same principle may be extended to powers of roots, when the exponents have a common denominator.
Thus a2/7.a3/7 = a2/7+3/7 =a5/7.
For the first numerator shows how often a1/7 is taken as a factor to produce a2/7. (Art. 246.)
And the second numerator shows how often a1/7 is taken as a factor to produce a3/7.
The sum of the numerators therefore, shows how often the root must be taken, for the product. (Art.100.)
Or thus, a2/7 = a1/7.a1/7.
And a3/7 = a1/7.a1/7.a1/7.
Therefore a2/7.a3/7 = a1/7.a1/7.a1/7.a1/7.a1/7= a5/7.
249. The value of a quantity is not altered, by applying to it a fractional index whose numerator and denominator are equal.
Thus a= 2/2 = a3/3=an/n. For the denominator shows that a is resolved into a certain number of factors; and the numerator shows that all these factors are included in an/n.
Thus a3/3=a1/3.a1/3.a1/3, which is equal to a.
And an/n = a1/n.a1/n..... n times.
On the other hand, when the numerator of a fractional index becomes equal to the denominator, the expression may be rendered more simple by rejecting the index.
Instead of an/n we may write a.
250. The index of a power or root may be exchanged, for any other index of the same value.
Instead of a2/3, we may put a4/6.
For in the latter of these expressions, a is supposed to be resolved into twice as many factors as in the former; and the numerator shows that twice as many of these factors are to be multiplied together. So that the whole value is not altered.
Thus x2/3 = x4/6 = x6/9. that is, the square of the cube root is the same, as the fourth power of the sixth root, the sixth power of the nin-th root.
So a2 = a4/2 = a6/3 = a2n/n. For the value of each of these indices is 2. (Art. 132.)
251. From the preceding article, it will be easily seen, that a fractional index may be expressed in decimals.
1. Thus a1/2 = a5/10, or a0,5; that is, the square root is equal to the 5th power of the ten-th root.
2. a1/4 = a25/100, or a0,25; that is, the fourth root is equal to the 25th power of the 100th root.
3. a2/5 = a0,4.
4. a7/2 = a3,5.
5. a9/5 = a1,8
In many cases, however, the decimal can be only an approximation to the true index.
Thus a1/3 =a0,3 nearly. a1/3 = a0,333334 very nearly.
In this manner, the approximation may be carried to any degree of exactness which is required.
Thus a5/3 = a1,66666. nbsp; a11/7 = a1,87142.
These decimal indices form a very important class of numbers, called logarithms.
It is frequently convenient to vary the notation of powers of roots, by making use of a vinculum, or the radical sign √. In doing this, we must keep in mind, that the power of a root is the same as the root of a power; (Art. 247,) and also, that the denominator of a fractional exponent expresses a root, and the numerator a power. (Art. 246.)
Instead, therefore, of a2/3 we may write (a1/3)2, or (a2)1/3, or 3√.
The first of these three forms denotes the square of the cube root of a; and each of the two last, the cube root of the square of a.
So am/n = (a1/n)m = (am)1/n = n√
And (bx)3/4 = (b3x3)1/4 = 4√ .
And (a + y)3/5 = [(a + y)3]1/5 = 5√ .