3. What is the 6th root of 481890304?

This may be done by extracting the 6th root directly, or by extracting first the second and then the third root.

done both ways.

4. What is the 7th root of 13492928512 ?

Let it be

XL. Fractional Exponents and Irrational Quantities.

The method explained above, Art. XXXVI, for extracting the roots of literal quantities, gives rise to fractional exponents, when they cannot be exactly divided by the number expressing the root. Since quantities of this kind frequently occur, mathematicians have invented methods of performing the different operations upon them in the same manner as if the roots could be found exactly; and thus putting off the actual extracting of the root until the last, if it happens to be most convenient. The expressions also may often be reduced to others much more simple, and whose roots may be more easily found. It has been already observed that the root of a quantity consisting of several factors, is the same as the product of the roots of the several factors.

Hence

་

We see that the same expression may be written in a great many different forms. The most remarkable of the above are

On this principle we may actually take the root of a part of the factors of a quantity when they have roots, and leave the roots of the others to be taken by approximation at a convenient time.

The quantity (72 a3 b3 c) may be resolved into factors thus. (2 × 36 aa a b1b c)2 = (36 a2 b1)*. (2 a b c)3.

The root of the first factor 36 a2 b can be found exactly, and the expression becomes

6 a b2 (2 a b c).

This expression is much more simple than the other, for now it is necessary to find the root of only 2 a b c.

The expression might have been put in this form,

(72)* a‡ b3 c1 = (36.2)1 a14 2a c = 6.2a a a‡b?b‡ c±

C

α

= 6 a b2 (2 a b c).

Examples.

1. Reduce (16 a3 b*) to its simplest form.

4. Reduce (16 a3 b3 +32 a2b3 m) to its simplest form.

(16 a3 b3 + 32 a2 b3 m)1 = (16 a2 b2)1 (a b3 + 2 b m) *

Sometimes it is convenient to multiply a root by another quantity, or one root by another.

If it is required to multiply (3 a' b) by a b, it may be expressed thus: a b (3 ab). But if it is required actually to unite them, a b must first be raised to the second power, and the pro

duct becomes (3 ab3). This will appear more plain in the following manner,

3a a ba × a b = 3a a2 b1‡ = 3a aa b* = (3 aa b3)*.

If instead of enclosing the quantity in the parenthesis and writing the exponent of the root over it, we divide the exponent of all the factors by the exponent of the root, all the operations will be very simple.

Let a be multiplied by a1.

X = a+++ =α.

a1 × a3 = a++1 = a* = a2.

aa b3 × aš b3 = a§+§ b3+‡ = a} b3.

3

That is, multiplication is performed on similar quantities by adding the exponents, as when the exponents are whole numbers. In like manner division is performed by subtracting the exponents.

It must be observed that a3 may be read, the third root of the second power of a, or the second power of the third root of a. For

Χα X = a + b + }

=

That is, a power of a root may be found by multiplying the fractional exponent by the exponent of the power.

Consequently a root of a root may be found by dividing the fractional index by the exponent of the root. In multiplying and dividing the fractional exponents, we must apply the same rules that we apply to common fractions.

The 3d root of a3 is a3.

The 3d root of až is aš.

The 5th root of a b is as b3%.

If the numerator and denominator both be multiplied or divided by the same number, the value of the quantity will not be altered; for that is the same as raising it to a power, and then extracting the root.

a = at

If it is required to multiply a

by a, the fractions may be reduced to a common denominator and added: thus,

The same may be done in division and the exponents subtracted.

In fact, quantities with fractional exponents are subject to precisely the same rules, as when the exponents are whole numbers; but the rules must be applied as to fractions. fractions may be reduced to decimals without altering the value; thus

The

It is very important to remember how these quantities may be separated into factors. Since multiplication is performed by adding the exponents, and division by subtracting them, any quantity may be separated into as many factors as we please, by separating the exponent into parts. Thus,

a3 = a3 X a2 = a × a1 = a X a2 × a2

The sum of all the exponents in the last expression is 5. Logarithms are of the same nature as these exponents, and afford as great a facility in operating upon numbers, as these do upon letters. And the operations are performed in the same way, as will be explained hereafter.

If the learner should ever have occasion to read other treatises on mathematics, he will generally find the roots expressed by what are called radical signs. The second root is expressed with the sign√, the third root✔ the same sign

with the index of the root over it. The 4th root is

4

&c.

They will be easily understood if the radical sign be removed, and the exponents divided by the index of the root or the quantity enclosed in a parenthesis, and the root written over it. The expression 5 a2 b3 becomes

4

The expression √ a2 + b2 is equivalent to (a2 + b2)&