This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — To evaluate the expression, we use the properties of the cube roots of unity, where 1 + + ^2 = 0. Step 1: Simplify the first term (1-+^2)^5. From the property 1 + + ^2 = 0, we know that 1 + ^2 = -. Substitute this into the first term: (1-+^2)^5 = ((1+^2) - )^5 = (- - )^5 = (-2)^5 Step 2: Simplify the second term (1+-^2)^5. From the property 1 + + ^2 = 0, we know that 1 + = -^2. Substitute this into the second term: (1+-^2)^5 = ((1+) - ^2)^5 = (-^2 - ^2)^5 = (-2^2)^5 Step 3: Add the simplified terms. (-2)^5 + (-2^2)^5 = (-2)^5 ^5 + (-2)^5 (^2)^5 = -32 ^5 - 32 ^10 Factor out -32: = -32 (^5 + ^10) Step 4: Simplify the powers of . We know that ^3 = 1. So, ^5 = ^3 · ^2 = 1 · ^2 = ^2. And ^10 = ^9 · = (^3)^3 · = 1^3 · = . Step 5: Substitute the simplified powers back into the expression. -32 (^2 + ) From the property 1 + + ^2 = 0, we know that + ^2 = -1. -32 (-1) = 32 The final answer is (d) 32. Drop the next question.