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ADDITION OF FRACTIONS.

41. In order that Algebraic Fractions may be added together, they must first be reduced to a common denominator. (No 52. Arith.)

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From Ex. 2. Case III, of Multiplication ao-b3=(a+b) (a—b);

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INVOLUTION AND EVOLUTION.

45. IT has been already stated, (No. 11.) that when a quantity is multiplied any number of times by itself, the successive products are called powers, of the original quantity, while this quantity itself, in reference to these powers, is denominated the root. Thus, the successive powers of a are a, aa, aaa, aaaa, &c. or according to the mode of notation explained (No. 11.) a1, a2, a3, a+, &c., where a2 and a3, from geometrical considerations, are respectively called the square and cube of the given quantity a. The fourth power of a or at, is also sometimes called the biquadrate; but the higher powers seem to have no other than that of numeral distinction. This process of continued multiplication of a quantity into itself, by which its successive powers are formed, is called Involution.

In the powers of Algebraic quantities, the multiplication is merely indicated, but in those of numbers, it is actually performed. Thus, the fifth power of r is expressed by xxxxx or more simply by x5, when there is a mere indication of the multiplication. The fifth power of 2 is = - 25, by indication, or 2×2×2×2×2=32 by actual multiplication. In the same manner, the fifth power of 3 is = 35 = 3x3x3x3x3 = 243, &c. when it must be observed, that the number of factors in the power, is that expressed by the units in its index. In this way the following table, which contains the first six powers of each of the nine digits is formed.

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5th 1 32 243 1024 3125 7776 16807 32768 59049

6th 1 64 729 4096 15625 46656 117649 262144 531441]

46. In raising a simple algebraic quantity a to any power, all that is necessary, is to multiply its index by the index of

the power to which it is to be raised.

Thus the 5th power

=

ofa is a 1X5 = a; the 4th power of 62 is 62 × 4 — 68. This is manifest, for b2 raised to the fourth power is = b2xb2xb2xb2 = b2+2+2+2 _ 2x4_68. In the same

=

manner, a" raised to the mth power is a" × m = am n.

The odd powers of negative quantities, will have the sign minus, and the even powers the sign plus. Thus, the square αχ -a= + a2; the cube of a=

of

αχ a = + a2 x

fourth power of

a3.

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Any power of the product of two or more quantities, is equal to the same power of each of the factors multiplied together: thus, (a b)2 = ab × ab=a2 b2 or in general (abc)m = am xbm x c' If the quantity have a numeral coefficient, this coefficient must likewise be raised to the given 2X4 3X4 power, thus (3 a2 b3)1 = 34 xa2×4 x 63×4 — 81a3b12. b

m.

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47. Fractions are raised to any given power, by raising both numerator and denominator to that given power. Thus,

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A mixed number or quantity is raised to any pow

er by first reducing it to an improper fraction, and then raising that fraction to the given power. Thus, the third power

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48. The powers of compound algebraic quantities are found by the mere application of the rule for the multiplication of compound quantities, (No. 26.)

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