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by the rules of multiplication, is +aa or +a”; but if we involve --a to the 3d power, we have (-a)x(-a)x(-a), which, by the same rules, is aaa or —as. It is evident that the 4th power of a is +a+; the 5th power of a is—as, &c.; the even powers being positive and the uneven powers negative. We shall thus find that (-a)20= +420, and (-a)"=-2%, &c. &c.
15. We see then that the squares of +a and -a are both +ar. If then it be required to extract the square root of +a', this root may be either +a or -a; so that the result is ambiguous, which is expressed by prefixing both signs thus, +a; that is, vas=ta.
But in extracting the cube root no ambiguity will exist, for the cube of +a and the cube of -a are not the same, the one being +aand the other -as; and, consequently, the cube root of ta3 is +a, and the cube root of -as is —a. In the same way the 4th root of +at will be either ta or -a, but the 5th root of tas will be +a, and the 5th root of -as will be
Hence the rule, the even root of a positive quantity is either positive or negative; the uneven root of a positive quantity is positive; and the uneven root of a negative quantity is negative.
16. The even root of a negative quantity is impossible. For the square root of -a® is neither ta nor -a; since ta squared is ta®, and a squared is +as; and there is no quantity which, multiplied by itself, will produce —a%. For the same reason TM-a, 9-25,–1 are impossible. These quantities, or rather expressions, are called imaginary. It is usual to reduce them as follows:
To find the square root of —a>, we have va®=VQ*X(-1)= Fav-1, (Arts. 11 and 15,) in which last form the imaginary part is the even root of -1 ; and to this form all imaginary expressions may be reduced. Thus,
d d ab ́c in which ✓
is a possible quantity, being the even root of a positive quantity.
1. Find the product of —x"Y"zľ, 2c"YPz, —2PYM?”.
Ans. xmtutrym+ntezminto or (Qyz)m+ntp. 2. Divide empynpzip by xnpy"PZMP. Ans.(am-ny"-429-m)". 3. Divide —axybaz by —amybus.
Ans.(aubs)-*. 4. Involve -b to the nth power.
Ans. +b" ifn is even, -on if n is uneven. 5. Involve — to the (2n)th power.
Ans. +ben (for 2n is even whatever be the value of n.) 6. Involve —b to the (2n+1)th power. Ans. —b2n+1 (for whatever be the value of n, 2n is even,
and 2n+1 is uneven.)
2xys 7. Find the 4th root of
1 8. Find the 3d root of
amn? 9. Find the 2d root of
Ans. + Xyy1.
a’xy 10. Find the nth root of amn.
Ans. tam if n is uneven, Eam if n is even. 11. Find the (2n)th root of—an.
Ans. Ea’ -1.
EXPONENTS IN GENERAL.
18. In the preceding chapter we have applied the rules for the multiplication, division, &c. of powers only to those cases in which the exponents were whole positive numbers. Exponents, however, may be also fractional and negative.
19. Let it be required to extract the 3d root of a?. We have seen that this would be var, but algebraists have agreed to apply to such cases the general rule for the evolution of powers given in Art. 9. To extract the 3d root of a?, according to this rule, we divide the exponent by 3, which gives aš ; therefore, vus and aš are equivalent expressions, both signifying “the 3d root of a squared;" and in the fractional form, the numerator of the fraction indicates the power, and the denominator the root. In the same way we find Va=a; Yazat; jao=aš; War=a*, &c.
The use of fractional exponents thus renders the radical sign unnecessary, and enables us to express roots and powers by the same general mode of notation.
20. Again, let it be required to divide as by a'. This is, which fraction reduced by dividing its numerator and denominator by as,
1 becomes But by Art. 6, powers of the same quantity are di.
a vided by subtracting their exponents. Hence, to divide as by a”, we must subtract 7 from 5; but 5—7=-2, so that the quotient
1 will be a with the exponent —2, or a-. Therefore,
and a-% are
a? equivalent expressions; but the latter has the advantage of representing a fraction in the same form with whole numbers. In the same way
1 we shall find
We see then that the negative exponent indicates the reciprocal of a power ; that is, a-is the reciprocal of a®; a-7 is the reciprocal of a?; a- is the reciprocal of an, &c.
21. We apply the rules given in the preceding chapter also to fractional and negative exponents. Thus, to multiply at by at we add the exponents (Art. 5,) and we have
a} xaš=a}+1=a. In the same manner we find,
22. Applying the rule for division of powers, (Art. 6,) we have
23. Applying the rule for involution of powers, (Art. 8,) we have
24. Applying the rule for evolution of powers, (Art. 9) we have
25. By using fractional exponents involution and evolution are performed by the same rule ; that is, by multiplying the exponents. Thus, the above examples will be performed,
Bat=(sažj} (Art. 19,) =stabx$ (Art. 8,)=247
26. We conclude this chapter by exhibiting the regular series of powers of a, decreasing by unity. Beginning at a" and dividing continually by a we have the following series :
am a', a», a-, ao, a 20, a-3 a-m+s, a-mt, a-m. which is a regular series of powers of a from +m to Any two terms equally distant from ao are the reciprocals of each other ; thus, al is the reciprocal of a1, a' of a , &c., and am of a The term ao is found (as the other terms are found) by subtract