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The Square Root of Numbers.

192. EXAMPLES.

1. Find the square root of 7 to 4 places of decimals.

Ans. 0·2425 +: 2. Find the square root of B to 3 places of decimals.

Ans. 0:645 + 3. Find the square root of 18 to 2 places of decimals.

Ans. 1.32+ 4. Find the square root of 111 to 3 places of decimals.

Ans. 3.418+ 5. Find the 3d root of to 3 places of decimals.

Ans. 0·873 + 6. Find the 3d root of to 3 places of decimals.

Ans. 0.941 + 7. Find the 3d root of 152 to 3 places of decimals.

Ans. 2.502 +.

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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 a 65 c
2. Find the mth power of an.
3. Find the — mth power of an.
4. Find the mth power of a-n.
5. Find the — mth power of a-.

Ans. 8 a6 615c3.

Ans. amn. Ans. a-m*. Ans. a-mn.

Ans. ama

Root of a Monomial; imaginary quantity.

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

Ans. a90 630 660. 7. Find the qth power of the — pth power of the mth power of a-n.

Ans. am n P 9. 8. Find the rth power of am banco d.

Ans. amo b-necpr dr. 9. Find the — 3d power of a-2b3c-5f6x-1.

Ans. abb-9c15f-18 23 a4 b5

a16 720 10. Find the 4th power of

Ans. c3df

c12d4f4

a2 33 11. Find the 2 mth power of the - 1st power of

cd5 a4 m 66 m Ans.

c m d 10 m 12. Find the 5th power of — 2 a?. Ans. 32 a10. 13. Find the 4th power of — 36-3. Ans 81 6-12. 14. Find the 5th power of the 4th power of the 3d power of

Ans. a60. 15. Find the 5th

power
of the 3d power of

Ans.

-a15.

m

a,

-a.

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

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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 f12 g-8n.

Ans. +3 a2bf-6 g 4". a8 620 c4

a2 b5 c 4. Find the 4th root of

Ans. + 16 d 12 z16°

40374 5. Find the 9th root of — 236 245 69. Ans. — 24 a5 b.

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Calculus of Radicals.

an 6P

n

8. Find the mth root of

cet Ans. am bmc 9. Find the 5th root of as.

Ans. 10. Find the square root of a.

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Ans. am.

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 ;

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