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 ao, or (ar) amp; for since, (Art. 274), (a")"=α2 Xarxa2, &c. ur+p+p+ etc.— apm. (1) 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 fuctor in the quantity by the exponent of the power. Thus the mth power of (abic), or (abic)m is = apmbamcm; for since (Art. 274), arbicr)=(a*bic)×(a2bc"), &c. = apa?... baba.. = (ao)m × (bo)m × (cr)m; by observing that in each of these products, such as aPap &c., or b b &c., there enter m equal factors. = (2); 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 a2•4 62.4=3X3×3×3×ab3 = 81a3b3. For the numerical coefficient is in this case the same as any other factor. 41664 256 1024 4096 5 25 125 625 3125 15625 78125 12. 12.08 16384 12. 1.587 2.236 1.71 1.817 1.913 279. Any power of a fraction is equal to the same power of the numerator divided by the like denominator. Thus the mth power of (Art. 274), () axaxa, etc. am bxbxb, etc. bm equal to m. power of the ; where the number of factors is a br And in like manner the mth power of or arba (αP)m(b¶)m___ αpmf. qm (c)(d) en drm cncr (3). 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)2m=[(i±a)3] " = ( + a2) m = + a2 m m 2 (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)em+1= (±a) 2 m × (±a) =a2m×±a=±a3ш+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, prefixing 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)=4Xa2 Xba=4a3b3. Ans. Xa2.3 xb2.3=-81ab6. Ans. (3a2b3)3-81 2 Ex. 3. What is the 4th power of —2a3 x2? Here, (-2a32)=(Art. 280), +(2a3 x2)=16 X a3. x2.416α12x2. Ans. Ex. 4. What is the cube, or third power of abc? Here, abc Xabc abc=a×a×a× b × b xbx c xc x c=α b3c3. 283. When the quantity to be involved is a fraction, raise both the numerator and denominator to the power proposed (Art. 279). Ex. 7. What is the 8th power of 2a2? Ex. 10. What is the 5th power of? Ans. Ex. 11. What is the 4th power of C 3125 5 Ex. 12. Required the cube of ? Ex. 13. Required the square of a2b2? Ans. a4b4. Ex. 14. Required the 9th power of -xy? Ans.xy. Ex. 15. Required the 0th power of xy? Ex. 16. Required the 4th power of a-2? Ans. 1. 1 8 CASE II. To involve a compound algebraic Quantity. RULE I. 284. Multiply the given quantity continually into itself as many times minus one as is denoted by the index of the power, as in the multiplication of com pound algebraic quantities (Art. 79), and the last product will be the power required. Ex. 1. What is the square of a+2b? a+26 a+26 a2+2ab +2ab+4b2 Square a2+4ab+4b2 Ex. 2. What is the cube of a2 —x2 ? a2 - x2 Ex. 3. Required the fourth power of a+3 Ans. a4+12a3b+54a2b2 +108ab3+81b4. Ex. 4. Required the square of 3x2+2x+5. Ans. 9x+12x3 +34x2+20x+25. Ex. 5. Required the cube of 3x-5. 3 Ans. 27x3-135x3+225x-125. Ex. 6. Required the cube of x2-2x+1. Ans. x-6x5+15x420x3+15x2-6x+1. Ex. 7. Required the fourth power of 2+3x. |
