in which a and b may represent the numerical coefficients of x and y. Now denote the numerical coefficients of the expansion by C1, C2, C3, etc. We shall then have (5a+3x)=625a +1500a*x+1350a*x+540ax+81x, Ans. EXAMPLES FOR PRACTICE. 1. Find the fourth power of 2x+5y. Ans. 16x+160x'y+600x'y'+1000.xy+625y. 2. Find the fifth power of 2a-3x. Ans. 32a-240a*x+720a'x-1080a'x'+810ax-243x. 3. Find the sixth power of 3+4x". Ans. 729+5832x+19440x + 34560x+34560x+18432x+ 4096x1. За 4r 4. Find the fourth power of +5 Ans.a+7 a3r+ §}a3r'+ }£}ar'+ §§§r*. Ans. + 33t°r+ 2°t*r3+20t3r3° + 135t3r* + 243tr°+ 729r®. 729 380. The approximate value of a surd root may be obtained with much facility by expanding the root into a series. Let a represent that perfect nth power, which is next less or next greater than the given number, and let b denote the difference between this power and the given number. Then a+b, or a"—b, will express the given number. But we have The second members of these equations contain no radicals; hence, Any surd may be developed into a series of rational terms; whence by summing the series, we may obtain approximately the indicated root. It should be observed that the smaller the fraction rapidly will the series converge. 1. Find the cube root of 76, to six decimal places. b is, the more The smallest fraction will result by taking the cube which is next less than 76, or 64; thus, We may now develop the radical part by equation (1), in which To form the binomial coefficients we have the factors, We represent the successive terms by A, B, C, etc.; and to secure accuracy in the final result to the 6th place of decimals, we should |