Root of a Monomial; imaginary quantity.

6. Find the 6th power of the 5th power of a3 b c2.

Ans. a90 b30 c60.

7. Find the qth power of the -pth power of the mth power of a-n.

8. Find the rth power of am b―n cP d.

9. Find the 3d power of a-2b3 c-5f6x-1.

Ans. amn pq.

Ans. amr b―nr cpr dr.

14. Find the 5th power of the 4th power of the 3d power

196. To find any root of a monomial.

Solution. Reversing the rule of art. 193, we obtain immediately the following rule.

Extract the required root of the coefficient; and divide each exponent by the exponent of the required

root.

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.

Ans. an.

2. Find the mth root of a-mn ̧

Ans. a-".

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

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.

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.

203. EXAMPLES.

3

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

3

7 54 a3 b5 c3 = 7 / 2. 33. a3 b5 c = 7. 2§. 3. a bễ c

=21.2.ab1+3c=21. 2a a b b3 o

= 21 abc2b2.

316 a3 b5c3 = 3√ 2a a3 b5 c = 3. 2a. a b‡ c

whence

= 3.2.2a a b b3 c = 6abc√2b3,

3

754a3 b3 c3+316 a3 b5 c3±21abc2b2+6 abc $2ba

=27 a b c 2 b2.

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.

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

Ans. Vabc.

5. Find the continued product of a ✔x, bŵy, c↓ z

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

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

Ans. ab

20

=ava

с

Ans. a2b-c= 66

13. Multiply 3+5 by 2-5. Ans. 1-5.

14. Multiply 7+26 by 9-5 √ 6.

17. Multiply —5—√‡ by −5+vz.

Ans. 241.