Final answer:
The student's expression for the binomial expansion to the fourth power contains typos and is therefore not correct. The correct expansion follows the pattern (a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4, with properly placed exponents.
Step-by-step explanation:
The student is asking whether the expansion (at-b)^4 is equal to a^4-4a^3b+6a^2b^2-4ab^3+b^4. It seems that there is a typographical error in the question. The correct expansion of a binomial to the fourth power is (a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4. The student's expression is missing the proper exponents in the second and fourth terms. Thus, the given expression is not an identity.
To clarify with an example, if a and b are real numbers, then (2 - 3)^4 is equal to 2^4 - 4(2^3)(3) + 6(2^2)(3^2) - 4(2)(3^3) + 3^4, which correctly simplifies to 16 - 96 + 108 - 108 + 81 = 1, demonstrating the correct expansion of the binomial to the fourth power.