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of the years of my age. What was the age of the father and son?

Ans. 53, and 18.

Prob. 36. Two persons, A and B, had a mind to purchase a house rated at 1200 dollars; says A to B, if you give me of your money, I can purchase the house alone; but, says B to A, if you will give me 2th of yours, I shall be able to purchase the house. How much money had each of them?

Ans. A had 800, and B 600 dollars. Prob. 37. There is a cistern into which water is admitted by three cocks, two of which are exactly of the same dimensions. When they are all open, fivetwelfths of the cistern is filled in 4 hours; and if one of the equal cocks be stopped, seven-ninths of the cistern is filled in 10 hours and 40 minutes. In how many hours would each cock fill the cistern?

Ans. Each of the equal ones in 32 hours, and the other in 24.

38. Two shepherds, A and B, are intrusted with the charge of two flocks of sheep. A's consisting chiefly of ewes, many of which produced lambs, is at the end of the year increased by 80; but B finds his stock diminished by 20; when their numbers are in the proportion of 8: 3. Now had A lost 20 of his sheep, and B had an increase of 90, the numbers would have been in the proportion of 7 to 10. What were the numbers ?

Ans. A's 160, and B's 110.

Prob. 39. At an election for two members of congress, three men offer themselves as candidates; the number of voters for the two successful ones are in the ratio of 9 to 8; and if the first had had 7 more, his majority over the second would have been to the majority of the second over the third as 12: 7. Now if the first and third had formed a coalition, and had one more voter, they would each have succeeded by a majority of 7: How many voted for each?

Ans. 369, 328, and 300, respectively,

CHAPTER VI.

ON

THE INVOLUTION AND EVOLUTION

OF NUMBERS, AND OF ALGEBRAIC QUANTITIES.

274. The powers of any quantity, are the successiveTM products, arising from unity, continually multiplied by that quantity. Or, the power of the order m of a quantity, m being a whole positive number, is the product of that quantity continually multiplied m times into itself, or till the number of factors amounts to the number of units in that given power.

C

-

1

275. INVOLUTION is the method of raising any quantity to a given power, EVOLUTION, or the extraction of roots, being just the reverse of Involution, is the method of determining a quantity which, raised to a proposed power, will produce a given quantity.

NOTE. The term root has been already defined, (Art 15).

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§ I. INVOLUTION OF ALGEBRAIC QUANTITIES.

276. It has been observed, (Art. 13), that the powers of algebraic quantities, are expressed by placing the index or exponent of the power over the quantity.

Hence, if a proposed root be a single letter and without a coefficient, any required power of it will be expresssed by the same letter with the index of the power written over it, Thus, the nth power of a isa"; u being any positive number whatever.

277. If the proposed root be itself a power, the required power will be obtained by multiplying the index of the given power into that of the required power. Thus the mth power of a2, or (ar)m=amp; for since, (Art. 274), (a3)"=α2 Xa2 Xa2, &c. =apte+p+ elc.— (1)

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where the number of factors ap is equal to m.

278. Also, if a simple quantity be composed of seve ral factors, it can be raised to any power by multiplying the index of every factor in the quantity by the exponent of the power. Thus the mth power of (abc"), or (arba) is = apmbqmcm; for since (Art. 274), arbicr)=(apbic") × (abic*), &c. apar... baba...

=(a3)m × (b¶)m × (cr)m; (2); by observing that in each of these products, such as a3а2 &c., or b ba &c., there enter m equal factors.

Cor. Hence, if the proposed quantity has a numerical coefficient, it must also be involved to the required power Thus the fourth power of 3a2b2 is = 3 aa•4 62.4=3×3× 3 × 3 × a® b3 =81ab3. For the numerical coefficient is in this case the same as any other factor.

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2187

1.732 1.442

16384

2.

1.587

78125

2.236

1.71 1.817

39 27 81 243 729

4 1664 256 1024 4096

5 25 125 625 3125 15625
6 36 216 1296 7776 46656 279936 2.449
7 49 343 2401 16807 117649 823543 2.646
8 64512 4096 32768 262144 20971522.828
981 729 656159049 531441 4782969|3.

1.913

2. 12.08

279. Any power of a fraction is equal to the same power of the numerator divided by the like power of the denominator.

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280. Any even power of a positive or negative quantity, is necessarily positive. In fact, 2m being the formula of even numbers, we have (±a)&m=[(1±a)3] "=(+a2)m=+a2m

(4).

281. Any odd power of a quantity will have the same sign as the quantity itself. For, the general formula of odd numbers, (Art. 111), being 2m+1, we have (a)2m+1=(a) am × (±a) =aam×±a=±22+1

(5).

2

The involution of algebraic quantities is generally divided into two cases.

CASE I.

To involve a simple algebraic Quantity.

RULE.

282. Raise the coefficient, if any, to the required power, then multiply the index of each factor, or letter, by the index of the required power, and write

their several products over their respective factors: Let the quantities thus arising be annexed to each other and to the same power of the coefficient, pre-. fixing the proper sign, and it will be the power required. Or, multiply the quantity into itself as many times less one as is denoted by the index of the power, and the last product, with the proper sign prefixed, will be the answer.

Ex. 1. Required the square, or second power of 2ab.

Here, (2ab)2=4Xa2 Xb2=4a2b2. Ans.

Ex. 2. What is the cube of -3a2b2 ? Here, (-3a2b2)3 (Art. 281), Xa2.3x62.3-81ab6. Ans.

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(3a2b2)3-31

Ex. 3. What is the 4th power of -2a3x2? Here, (-2a33)= (Art. 280), +(2a3x2)=16 X a3• x2+4= 16a12x8. Ans.

Ex. 4. What is the cube, or third power of abe? Here, abc Xabc Xabc=a XaXaxbxbxbxcxcx

283. When the quantity to be involved is a fraction, raise both the numerator and denominator to the power proposed (Art. 279).

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