Using the definition of a root, express c as the root of the second member in each of the following equations:

17. x2-b. 18. x3-b2. 19. x1-b3. 20. x-b1. 21. x"=b". as a root of the first member of each of the

22-26. Express

equations in 18-21.

THEOREMS IN EVOLUTION

In any case evolution is merely a special case of factoring, in which all the factors are equal. That is, the square root, the cube root, the fourth root, etc., are found by taking one of two, of three, of four, etc., equal factors, respectively of the given expression.

Since even roots of negative numbers are not considered in this chapter and since the odd root of a negative number can be found by taking the like root of the same positive number (2277, 3), methods and rules for finding principal roots of positive numbers and expressions only will now be given.

It is to be assumed in what follows that the radicand is a positive number or quantity and that the roots taken are principal roots.

279. THEOREM I.-The nth root of the product of several positive factors is equal to the product of the nth roots of each factor. Thus, "vā. "vō · "V c.

"Vabe

=

a

"ν

(1) For, by the definition of the nth root,

("abe)" abe, and

=

3

3.

3√ — 27 ab3x3 = v−27. Và vào và

3

3

3.

280. THEOREM II.-The nth root of the quotient of two quantities is equal to the quotient of the nth roots of the dividend and divisor. Thus,

281. THEOREM III. To raise a radical to the nth power it is sufficient to raise the quantity under the radical to the nth power.

to n factors, and, ac

For ("Va)"=("1 a) ("v@) ("vā)

cording to Theorem I, 279, the product of the n radicals, each

equal to "a, is equal to "a · a · a

.

to n factors

=

"Va".

282. THEOREM IV.-In order to extract the nth root of a radical, the index of the radical is multiplied by n.

For, in order that a number may be raised to the mnth power, it can first be raised to the nth power and the result to the mth power. This would give a for the mnth power of the first member, and a for the mnth power of the second member, by definition of a root. Or, the nth power of "V"Va is "va; the power of "a therefore, the math power of "1 "a is a; and consequently,

Thus, for example,

3

m

mth

1 31 64=°1 64°1 2o=2

V 729 a1 3° (a) x 3 a2x.

a is a;

283. THEOREM V.-The arithmetical value of a radical is not changed by multiplying, or dividing the index of the radical and the exponent under the radical by the same positive integer.

Let p be any positive number; then

пр

np.

"Vam = "P√ amp.

[Th IV; 2282]

For, "PV/amp="V"↓ (aTM)"="√/a".

284. In case the root of a number or quantity is not readily detected, it may be found by resolving the number into its prime factors. Thus, to find the square root of 3111696 and the square root of the result:

Simplify the following examples by means of the preceding theorems and principles:

3

1. Vab, Væ, V9ab2, V-27 ab3%, V/81 x1y*z*,

285. The Square Root of Compound Quantities. The extraction of square roots of numbers in Arithmetic is based upon the method for finding the square root of a compound algebraic expression. This method will now be explained.

Since it is known that the square root of a2 + 2 ab + b2 is a + b, a general rule may be deduced for finding the square root of an algebraic expression by observing in what manner a + b is derived from a2+2ab+b2; thus,

Arrange the expression according to the powers of a; then the first term is a2, and its square root is a, which is the first term of the required root. Subtract the square of the term of the root just found, namely a2, from the expression and bring down the remainder, 2 ab+b2. Take twice the part of the root already found, namely 2a, for the first term of the new trial divisor, and divide the first term of the remainder, namely, 2 ab, by 2 a, obtaining a quotient b, the second term of the root; annex this to the first term of the trial divisor 2 a, obtaining 2a + b as a complete divisor and multiply it by b, the second term of the root; this gives 2ab+b2, which subtracted from the remainder leaves zero. This completes the operation in this case. If there were more than three terms in the expression, then the process with a+b would be like that with a.

At the be

Up to the third step the process is the same as above. ginning of the third step the root already found, namely, a + b, is doubled, 2a 2b being obtained for the first part of the trial divisor. Divide the first two terms of the remainder by it, obtaining -2 c; this is the third term in the root; annex this term to the first two terms in the trial divisor for a complete trial divisor and multiply the sum by the third term of the root, i. e., (2 a + 2b − 2 c) (— 2 c), and subtract the result from the remainder. In this case the operation is now complete. In case more terms are left in the remainder after the third step, the process must be continued till the square root is found.

286. EXAMPLES. The method just explained may be extended to expressions of more terms, if care is taken to obtain the trial divisor at each step of the process, by doubling the part of the root already found, and to obtain the complete divisor by annexing the new term of the root to the trial divisor.

8

I. Find the square root of x+25.x2+10x1—4x3 —20 x3+16-24x. Arrange the terms in the ascending powers of x; thus:

- 3 x

16-24x+25x-20 x3+10x-4x+x64-3x+2x2-x3

Here the square root of 16 is 4; this is the first term of the root. Subtract 16, the square of 4, from the whole expression; and the remainder is 24x+25 x2 20 x3 10 x .— 4 x3 + x3. Divide -24x by twice 4, or 8, obtaining

3.x, the second term of the root,