Question - Simplified Evaluation of Complex Number Expression
Solution:
The expression you're being asked to evaluate is:(1 + 3i)^8 + (1 - 3i)^8This is an expression involving complex numbers. When raised to powers, complex numbers can sometimes simplify due to their periodic nature in the complex plane, but it is usually not feasible to compute high powers like 8 by hand without a significant amount of computation. In this case, there's a shortcut that can simplify the process using binomial expansion and the fact that i^2 = -1.To evaluate this expression, you would normally apply the binomial theorem, which states that:(a + b)^n = Σ[k=0 to n] (n choose k) * a^(n-k) * b^kWhere "n choose k" is the binomial coefficient, calculated as:(n choose k) = n! / (k! * (n - k)!)However, for the powers of 8, this would involve a lot of terms and computation, namely:(1 + 3i)^8 = 1^8 + (8 choose 1)*1^7*(3i) + (8 choose 2)*1^6*(3i)^2 + ... + (3i)^8(1 - 3i)^8 = 1^8 + (8 choose 1)*1^7*(-3i) + (8 choose 2)*1^6*(-3i)^2 + ... + (-3i)^8You will notice that terms with odd powers of (3i) in (1 + 3i)^8 will cancel out with the corresponding terms in (1 - 3i)^8 due to the opposite signs. Only the even powers will remain, which are identical in both expressions because (-3i)^2n = (3i)^2n for all integers n, as both will be a real number.To simplify calculation, you can just compute the terms involving even powers of i from either (1 + 3i)^8 or (1 - 3i)^8 and then double the real part to get the final answer. However, for powers of 8 and without further tricks, this could be a tedious process and is typically done using a computer algebra system.For simplicity and practicality, let me assist you in solving this with such computational help:(1 + 3i)^8 + (1 - 3i)^8 simplifies to 2 * (1^8 + (8 choose 2)*1^6*(3i)^2 + (8 choose 4)*1^4*(3i)^4 + (8 choose 6)*1^2*(3i)^6 + (3i)^8).Each individual term can be computed, keeping in mind that i^2 = -1, i^4 = i^2 * i^2 = (-1)^2 = 1, i^6 = i^4 * i^2 = 1 * (-1) = -1, and i^8 = i^6 * i^2 = (-1) * (-1) = 1.Here's the simplification of the even-i-powered terms:(8 choose 2)(3i)^2 = 28 * 9 * (-1) = -252(8 choose 4)(3i)^4 = 70 * 81 * 1 = 5670(8 choose 6)(3i)^6 = 28 * 729 * (-1) = -20412(3i)^8 = 6561 * 1 = 6561Summing these and multiplying by 2 gives the real part:2 * (1 - 252 + 5670 - 20412 + 6561) = 2 * (1 - 252 + 5670 - 20412 + 6561) = 2 * 51568 = 103136So the final answer is:103136
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