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Thus, the square root of 9 is written “V9 or V9– 3; the fourth root of 16 is written ‘i’ 16= 2; and the n” root of a is written "Va. Here the indices of the roots are respectively 2, 4, and n.
274. A parenthesis, or vinculum, is often used to express the root of a quantity consisting of more than one term. Thus 1/16+25 means the sum of V16 and 25, while V16+25 means the square root of the sum of 16 and 25. Moreover, "Vo. 9" means the product of y” and the cube root of wo, while "1/oxy' means the cube root of the product a "y”.
Parentheses are sometimes used instead of the vinculum in connection with the radical sign. Thus, the same result may be ex
pressed by 1/16-L 25 or 1/(16 + 25).
275. Like and Unlike Roots.—Two roots are said to be like or unlike according as the indices of the roots are equal or unequal, whether the quantities under the radical sign are equal or not. Thus,
276. In this chapter will be considered the roots of numbers which are powers whose exponents are multiples of the indices of the roots.
An even root of a number is one whose index is an even number; thus
1. Any even root of a positive number will have the double sign =E; because either a positive or a negative number raised to an even power is positive, £262. Thus,
3. An odd root of a negative number is minus the same root of a number which has the same absolute value. Thus,
Hence, to find an odd root of a negative number, find the same root of the positive number which has the same absolute value, and prefia, the negative sign to this root.
4. Since 0° = 0, therefore V0 = 0. In general, since 0" = 0, ... "W/0 = 0. 5. The even root of a negative number can not be taken; because no
real number raised to an even power can produce a negative number. Such roots are called impossible. Thus,
1/- 5 can not be +3 or –3, since (+3)* = 9 and(–3)* = 9. 1/T2 can not be + æ or — ar, since (+ æ)* = c and (–2)* = wo. *1/–o can not be + a or — a, since (+ a)* = a” and (— a)* = a”. Even roots of negative numbers can not be expressed in terms of numbers hitherto used, i.e., in terms of positive or negative integers, positive or negative fractions, or of positive or negative roots that can be found. The roots of numbers which are not powers with exponents which are multiples of the indices of the required roots and even roots of negative numbers will be discussed later. REMARK.—It has been shown above that a positive number which is the nth power of It will be shown that any number has two square roots, three cube roots, four fourth roots, and five fifth roots; and in general it may be proved that any number has n, n” roots. Using the definition of a root, express c as the root of the second member in each of the following equations: 17. ac”—b. 18. ar"—b”. 19. ac"—b°. 20, ac"—l,7. 21. ac"—b". 22–26. Express b as a root of the first member of each of the equations in 18–21.
a number has at least one nth root and, when n is even, at least two; also that any negative number which is an odd power of a negative number has at least one odd root.
278. Principal Root.—1. The principal root of a positive number is its one positive root. Thus, 3 is the principal square root of 9, and 6 is the principal cube root of 216.
2. The principal odd root of a negative number is its one negative root.
Thus, -3 is the principal cube root of — 27, and — a the principal (2n + 1)" root of – a”.
3. It should be noticed at this point that the relation,
"1/an - ("va)" holds for the principal * root only. For, by the preceding article, the "Va" has n values, the principal value being a. But, by the definition of a root, ("1/a)" = a for every no root of a. Thus, W5° = + 5, if the negative root – 5, as well as the principal root + 5, is admitted; but 1/5 = (1/5)
in case of the principal square root only. In the work which follows, the radical sign will be used to repre
sent the principal root only, unless the contrary is expressly stated.
Find one cube root of: 5. 125. 6. – 216. 7. — 1000. 8. — ao".
Find the values of the principal roots indicated in the following examples:
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 (4277, 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.
281. THEOREM III. — To raise a radical to the no power it is sufficient to raise the quantity under the radical to the no power. Thus, ("1/a)"= "lo. For ("1/a)"=("I a) ("I'd) ("I a) . . . to n factors, and, according to Theorem I, #279, the product of the n radicals, each
Thus, - - (‘V16)* = ‘l 16 = ‘i (4) = 1 + = 4 (N;) - 'NG; "— 'No. a"c" 4 a.o." 27 l,6 27 #) 729, to T 9 to 282. THEOREM IV.-In order to eactract the n" root of a radical, the inder of the radical is multiplied by n. Thus, "1. "Wa- "1/a. For, in order that a number may be raised to the mn" power, it can first be raised to the n” power and the result to the m”.power. This would give a for the mn" power of the first member, and a for the mn" power of the second member, by definition of a root. Or, the no power of "V"va is "I as the m" power of "Va is a: "I'm
therefore, the mn" power of a is a ; and consequently,
Thus, for example, 1 * 64–" 64="1 2–2 *117 729 go-", 37(o– 3 a”a. 283. THEoREM. W. — 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