19. What is the fourth power of -10cd'x'y'? Ans. 20. What is the fifth power of —3a2b3x3? PROBLEM III. Ans. (124.) To raise a fraction to any power. A fraction is raised to a given power by multiplying the fraction by itself; that is, by multiplying the numerator by itself and the denominator by itself, Art. 85. QUEST.-How do we raise a fraction to any power" (125.) To raise a polynomial to any power. A polynomial is raised to a given power by multi plying the quantity continually by itself. Ex. 1. Let it be required to find the fourth power of a+b. a+b a2 + a b +ab+b2 (a+b)'=a*+2ab+b', the second power of a+b. a + b a'+2ab+ ab (a+b)'=a'+3a'b+3ab'+b', the third power. a + b a'+3a b+3a'b'+ ab3 +ab+3a2b2+3ab3+b* (a+b)'=a'+4a'b+6a2b'+4ab'+b', the fourth power. QUEST.-How do we raise a polynomial to any power? (126.) It will be observed that the number of multiplications is one less than the exponent of the power. Thus, to obtain the second power, we multiply the quantity by itself once; to obtain the third power, we multiply the quantity by itself twice, etc. Exponents may be applied to polynomials as well as to monomials. Thus (a+b)* denotes the fourth power of the expression a+b. 2. Find the fourth power of the binomial a-b. (a−b)'=a2-2ab+b2, the second power of a-b. (a—b)'=a3-3a'b+3a b2-b3, the third power. (a—b)*=a*—4a3b+6a2b3—4ab3+b', the fourth power. 3. What is the cube of 2a-1? Ans. 8a-12a2+6a-1. 4. What is the square of a+b+c? Ans. a+b+c2+2ab+2ac+2bc. QUEST.-How does the number of multiplications compare with the exponent of the power? 5. What is the square of 2a3-3b1? Ans. 4a-12a2b2+9b* 6. What is the cube of 2a-3b? Ans. 8a-36a'b+54ab2-2763. 7. What is the cube of 2ab+cd? Ans. 8a'b'+12a b'cd+6abc'd'+c3d3. 8. What is the fourth power of 3a-b? Ans. 81a-108a b+54a2b2—12ab3+b*. 9. What is the cube of a+b+c? Ans. a+b+c+3ab'+3ac2+3a2b+3a2c+3bc2+ 3b'c+6abc. 10. What is the cube of 2a-2ab+b2? BINOMIAL THEOREM. Ans. (127.) By the method already explained, any power of a binomial may be obtained by actual multiplication; but by the binomial theorem this labor may be greatly abridged. The successive powers of the binomial a+b, found by actual multiplication, are as follows: (a+b)'=a+b (a+b)'=a'+2ab+b2 (a+b)'=a'+3a2b+ 3a b2+b3 (a+b)*=a*+4a3b+ 6a2b2+ 4a b3+b* (a+b)'=a+5a'b+10a b'+10a'b'+ 5a b'+b° (a+b)'=a®+6ab+15a*b2+20a3b3+15a3b*+6ab®+b'. The powers of a-b, found in the same manner, are as follows: QUEST.-What is the object of the binomial theorem? (a-b)'=a-b (a-b)'=a2-2ab+b2 (a-b)3=a3-3a2b+ 3a b2—b3 (a-b)'=a*-4a'b+ 6a'b'- 4a b3+b' (a—b)'=a3—5a1b+10a3b3—10a3b3+5ab1—b3 (a−b)'=a®-6a'b+15a*b*—20a3b3+15a2b*—6ab'+bo. (128.) On comparing the powers of a+b with those of a-b, we perceive that they only differ in the signs of certain terms. In the powers of a+b, all the terms containing the odd powers of b have the sign —, while the even powers retain the sign +. The reason of this is obvious; for, since -b is the only negative term of the root, the terms of the power can only be rendered negative by b. A term which contains the factor -b an even number of times, will, therefore, be positive; a term which contains -b an odd number of times, must be negative. Hence we conclude, 1st. When both terms of the binomial are positive, all the terms of the power are positive. 2d. When the second term of the binomial is negative, all the odd terms of each power, counted from the left, are positive, and all the even terms negative. (129.) Also, we perceive that the number of terms in the power is always greater by unity than the exponent of the power. Thus the second power of the binomial contains three terms, the third power contains four terms, the fourth power contains five terms, etc. QUEST.-How do the powers of a+b compare with those of a-b? What terms will be positive and what will be negative? What is the number of terms in the power? |