Simplify (1+ω2)5(1 + \omega^2)^5.

Answer: −ω2-\omega^2

Step-by-step solution

Step 1: Use the property of cube roots of unity

Let ω\omega denote a non-real complex cube root of unity. The sum of the cube roots of unity satisfies 1+ω+ω2=01 + \omega + \omega^2 = 0, which gives 1+ω2=−ω1 + \omega^2 = -\omega.

Step 2: Substitute into the given expression

We replace 1+ω21 + \omega^2 with −ω-\omega in the given expression. Raising −ω-\omega to the power 55 gives (−1)5ω5=−ω5(-1)^5 \omega^5 = -\omega^5.

Step 3: Simplify using higher powers of omega

Since ω3=1\omega^3 = 1, we can express ω5\omega^5 as ω3⋅ω2=1⋅ω2=ω2\omega^3 \cdot \omega^2 = 1 \cdot \omega^2 = \omega^2. Therefore, the simplified value is −ω2-\omega^2.

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