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Fractional Exponents; imaginary quantities.

197. Corollary. The odd root of a positive quantity is, by art. 194, positive, and that of a negative quantity is negative. The even root of a positive quantity may be either positive or negative, which is expressed by the double sign ± preceding it. But, since the even powers of all quantities, whether positive or negative, are positive, the even root of a negative quantity can be neither a positive quantity nor a negative quantity, and it is, as it is called, an imaginary quantity.

198. Corollary. When the exponent of a letter is not exactly divisible by the exponent of the root to be extracted, a fractional exponent is obtained, which may consequently be used to represent the radical sign.

199. EXAMPLES.

1. Find the mth root of amn.

2. Find the mth root of a―mn.

Ans. an.

Ans. a-".

-8

3. Find the square root of 9 a4 b2 f-12 g−8 n ̧

Ans. 3 a2bf-6 g-4.

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200. Corollary. By taking out -1 as the factor of a negative quantity, of which an even root is to be extracted, the root of each factor may be extracted separately.

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202. Most of the difficulties in the calculation of radical quantities will be found to disappear if fractional exponents are substituted for the radical signs, and if the rules, before given for exponents, are applied to fractional exponents..

In the results thus obtained, radical signs may again be substituted for the fractional exponents ;

Examples in the Calculus of Radical Quantities.

but, before this substitution is made, the fractional exponents in each term should be reduced to a common denominator, in order that one radical sign may be sufficient for each term.

When numbers occur under the radical sign, they should be separated into their factors, and the roots of these factors should be extracted as far as pos

sible.

Fractional exponents greater than unity should often be reduced to mixed numbers.

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1. Add together 754 a3 65 c3 and 3 † 16 a3 b5 c3.

Solution. We have

754 a3 b5 c3 = 72.33. a3 b5 c = 7.2§. 3. a ¿§ c

3

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=21.2}.ab1+ c = 21. 2a a b b3 c

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3 † 16 a3 b5 c3 = 3√ 24 a3 b5 c = 3. 2$. a b‡ c

whence

3

= 3. 2. 2a a b b3 c = 6 a b c † 2 b2,

3

c=6abc2

754a3 b5c3+3 16 a3 b5c3=21abc2b2+6abcŶ2b2

=27 a b c 2 b2.
abc2b2.

2. From the sum of ✔✔✅24 and ✔✅54 subtract √6.

Ans. 46.

3. From the sum of ✅✔✅45 c3 and ✅5a2c subtract 80 c3.

Ans. (a-c)5c.

Examples in the Calculus of Radical Quantities.

m

m

m

4. Find the continued product of a, b, and ✔c.

m

Ans. ✔abc.

n

n

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5. Find the continued product of a√x, b y, c √ z.

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10. Find the continued product of a, a, a-t.

11. Multiply 6-2a-3 by a bbc.

-,a,

20

Ans. aaa.

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17. Multiply-5-✓ by -5+√2.

Ans. 241.

Examples in the Calculus of Radical Quantities.

18. Multiply 9+2 10 by 9-210.

Ans. 41.

19. Multiply 28+3√5—7√ 2 by ✔ 72—5 √/20 Ans. 1744210.

-22.

20. Multiply a+b by a√b.

Ans. a2-b.

21. Multiply a+b by va-b. Ans. a-b.

22. Multiply ✔a+c↓b by √a—c.
a-cb.

4

5

23. Multiply a3+ b2 by ŵ a3 — $/b2.

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Ans. aa_c2 b2. c2b2.

Ans. aab.

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