Given: binomial expansion: #(ax+by)^7# ; 1st coefficient #= 127#;
2nd coefficient #= -224#
Binomial expansion for #(ax+by)^7:#
#= _7C_0 (ax)^7(by)^0 + _7C_1 (ax)^6 (by)^1 + _7C_2(ax)^5(by)^2 + ... +_7C_7(ax)^0(by)^7#
#= _7C_0 a^7x^7 + _7C_1 a^6x^6 by + _7C_2 a^5x^5b^2y^2 + ... +_7C_7 b^7y^7#
Combinations:
# " "_7C_0 = (7!)/((7-0)!0!) = (7!)/(7!) = 1 = _7C_7#
#" "_7C_1 = (7!)/((7-1)!1!) = (7!)/(6!) = (7*6!)/(6!) = 7#
#" "_7C_2 = (7!)/((7-2)!2!) = (7!)/(5!2*1) = (7*6*5!)/(5!*2) = 21#
Binomial expansion for #(ax+by)^7:#
#= a^7x^7 + 7 a^6b x^6 y + 21 a^5b^2x^5y^2 + ... + b^7y^7#
Using the given coefficients for the first two terms, find #a " & "b#:
#a^7 = 127; " " 7 a^6b = -224#
#a = root(7)(127) = (127)^(1/7)#
#b = -224/(7a^6) = -224/(7 (root(7)(127))^6) = -32/(127)^(6/7)*(root(7)(127))/(root(7)(127)) #
#b= (-32root(7)(127))/127#
