281. THEOREM III. To raise a radical to the nth

power it is suffi

cient to raise the quantity under the radical to the nth power.

Thus,

("V ́@)" = "v/@".

For ("1/a)”=("1 a) ("v ́@) ("va)

to n factors, and, ac

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

equal to "a, is equal to "y ́a · a · a.

to n factors = TM V ́a”.

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

For, in order that a number may be raised to the math 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"a is "I therefore, the mnth

Thus, for example,

3

power

of "a is a;

a;

of power

m

the mth I a a is a;

and consequently,

1 31 64=°1 64=°1_2′′=2

1/ 729 a1x =°1′ 3o(a2)°x® = 3 a2x.

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.

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:

16.

1 (3⁄4V ́64«x°yTM × °V ́(45,x1oy15) × "V/3"x4y2n2–în).

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+b; thus,

a2 + 2 ab + b2 [a + b = root

a2

2a+b 2 ab + b2

2 ab + b2

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 a", from the expression and bring down the remainder, 2ab+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 2a, 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.

Thus, find the square root of a2 + 2 ab + b2 — 4 ac

- 4 be + 4 c2.

Up to the third step the process is the same as above.

At the be

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.

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

8-3x

16—24.x+25 x2—20 x3+10 x1—4 x3+x° | 4—3 x+2x2—x3

16

−24 x+25 x2—20 x3 +10 x31—4x2 + x®

-24x+9x2

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 x1. 4x2 + xo. Divide -24x by twice 4, or 8, obtaining 3r, the second term of the root,

and annex it to 8, the first term in the trial divisor; multiply the result by 3x and subtract the product from the remainder, leaving a second remainder, 162 - 20 x3 + 10 x1 4x+x. Double the root already found, obtaining 86x, the first part of the second trial divisor; divide 16x by 8, obtaining +22 for the third term in the root; annex this term to the trial divisor 8 6x, multiply the sum 8 6x+2x2 by 2x2 and subtract the product from the second remainder; this operation gives a third remainder, 8x+6x — 4x+x. Double the root already found, obtaining 8-6x+4x2 for the first part of the third trial divisor; then divide -8 x3 by 8, obtaining 3 as the fourth term of the root. Now from the above process there is no remainder on the completion of the last step. The operation is completed in this case.

This problem could have been solved with the same ease by arranging the expression with respect to the descending powers of x. II. Find the square root of

9 x2-6xb+30 xc +6 xd + b2 10bc2bd25 c2 + 10 cd + d2. Arrange the terms with respect to x thus:

9x2-6xb+30 xc+6.xd+b2-10bc-2bd+25c+10cd+d 3x-b+5c+d

It will be noticed that each trial divisor is equal to the preceding

with the last term doubled.

287. The fourth root of an expression can be found by extracting the square root of the square root. Similarly, the eighth root may be found by extracting the square root three successive times; and the sixteenth root by four successive extractions of the square root, and so on (see (282).

For example, find the fourth root of

81 x1 432 x3 + 864 x2

768x+256.

By proceeding as in the foregoing example the square root of the proposed expression will be found to be 9 x 24 x + 16; and the 9x2. square root of this is 3 x 4, which is the fourth root of the given

expression.