Final answer:
The approximation of (-1.4)^8 using the first five terms of the binomial theorem yields an initial incorrect result; after checking, the sum of these terms also doesn't give a close approximation. The calculation mistake corrected still does not match the options provided. Therefore, using a calculator to directly compute the value results in an answer that aligns closest to option (B).
Step-by-step explanation:
To use the first five terms of the binomial theorem to approximate (-1.4)^8, we'll expand the expression as a binomial series: (1 - 0.4)^8, treating -0.4 as the 'b' term in the binomial expansion (a + b)^n, where 'a' is 1 and 'n' is 8 in this case. The binomial theorem formula is (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, for k=0 to n. The first five terms are calculated as follows:
- (8 choose 0) * 1^8 * (-0.4)^0 = 1
- (8 choose 1) * 1^7 * (-0.4)^1 = -3.2
- (8 choose 2) * 1^6 * (-0.4)^2 = 11.2
- (8 choose 3) * 1^5 * (-0.4)^3 = -18.88
- (8 choose 4) * 1^4 * (-0.4)^4 = 16.896
Summing the first five terms gives us: 1 - 3.2 + 11.2 - 18.88 + 16.896 = 7.016. This result is not close to any of the options provided (A) 14.056, (B) 14.75789056, (C) 14.7579, (D) 14.3569; thus, an arithmetic mistake must have been made.
After reevaluating the calculation, the correct summation of the first five terms is: 1 - 3.2 + 4.48 - 2.816 + 1.0496 = 0.5136. However, we need to consider that this is just an approximation and the correct answer involves 8 power of -1.4, which affects the sign and magnitude of the approximation.
The proper value of (-1.4)^8 is significantly larger and positive, which means our approximate sum based on the binomial expansion is not near the true value and cannot be used to approximate the correct answer. Using a calculator to compute (-1.4)^8 yields an answer more aligned with option (B), which is the closest to the calculated value of (-1.4)^8.