Power of a Monomial.

CHAPTER V.

POWERS AND ROOTS.

SECTION I.

Powers and Roots of Monomials.

193. Problem. To find any power of a monomial.

Solution. The rule of art. 28, applied to this case, in which the factors are all equal, gives for the coefficient of the required power the same power of the given coefficient, and for the exponent of each letter the given exponent added to itself as many times as there are units in the exponent of the required power. Hence

Raise the coefficient of the given monomial to the required power; and multiply each exponent by the exponent of the required power.

194. Corollary. An even power of a negative quantity is, by art. 32, positive, and an odd power is negative.

195. EXAMPLES.

1. Find the third power of 2 a2 65 c.
2. Find the mth power of a".
3. Find the - mth power of a".

4. Find the mth power of a-n.
5. Find the - mth power of a-".

Ans. 8 a6 615 с3.

Ans. am n. Ans. a mn Ans.a-mn. Ans. amn.

•

Root of a Monomial; imaginary quantity.

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

Ans. a90 630 €60. 7. Find the qth power of the - pth power of the mth power of a-". Ans. amnpq.

8. Find the rth power of amb-n cr d.

Ans. amrb-nr cpr dr.

9. Find the 3d power of a-2b3c-5 f6 x-1.

10. Find the 4th power of

Ans. ab-9 c15 f-18 23. a4 b5 α16 620 Ans. cdf c12 d4 f4 a2 b3 cd5

11. Find the 2 mth power of the - 1st power of

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

of-a.

Ans. a60.

15. Find the - 5th power of the - 3d power of - a.

a

16. Find the - 4th power of the - 3d power of

Ans.

196. To find any root of a monomial.

a 12 612

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

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. a". 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 fractionalexponents 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 possible.

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

203. EXAMPLES.

3

3

1. Add together 754 3 65 c3 and 316 3 65 3

Solution. We have

3

754 a3b5c3=7ン 2.33. a3 65 c = 7.23.3.abc

= 21.23. ab + c = 21. 23 ab bc

3/16 a3b5c3 = 3√24 a3 b5 c = 3 . 24 . a b c

3. From the sum of

5 a2c subtract 80 c3. Ans. (a-c)√5c.