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
To evaluate the given product, we will use the properties of the cube roots of unity: 1. 1 + + ^2 = 0 2. ^3 = 1 The product is given as (1-+^2)(1-^2+^4)(1-^4+^8)... to 2n factors. Step 1: Simplify the first factor (1-+^2). Using the property 1++^2=0, we can write 1+^2 = -. (1-+^2) = (1+^2) - = - - = -2 Step 2: Simplify the second factor (1-^2+^4). Using the property ^3=1, we have ^4 = ^3 · = 1 · = . So the second factor becomes (1-^2+). Using the property 1++^2=0, we can write 1+ = -^2. (1-^2+) = (1+) - ^2 = -^2 - ^2 = -2^2 Step 3: Simplify the third factor (1-^4+^8). Using the property ^3=1, we have ^4 = and ^8 = ^6 · ^2 = (^3)^2 · ^2 = 1^2 · ^2 = ^2. So the third factor becomes (1-+^2). This is the same as the first factor. Step 4: Identify the pattern of the factors. The factors repeat in a cycle of two: (-2) and (-2^2). Since there are 2n factors in total, there will be n pairs of these two factors. Step 5: Calculate the product of 2n factors. The product P is given by: P = [(-2) · (-2^2)]^n P = [4 ^3]^n Using the property ^3=1: P = [4 · 1]^n P = 4^n Step 6: Express the result in terms of powers of 2. 4^n = (2^2)^n = 2^2n Comparing this with the given options, it matches option (b). The final answer is (b) 2^2n. Last free one today — make it count tomorrow, or type /upgrade for unlimited.