A 2025 Special | Problem 471

[@scottleung9587]

Cool - happy 2025!

[@aplusbi]

You, too!

[@scottleung9587]

@@aplusbi hopefully this year will be lucky - it's the only "perfect square" year in our lifetime.

[@mcwulf25]

A long way round to show |z| = 1. Have a very happy 2025 Syber.

[@aplusbi]

Thanks! You, too

[@calculusmethods]

Good explanation ❤

[@aplusbi]

Thanks! 😁

[@Don-Ensley]

problem z ²⁰²⁴ = z̅ Multiply both sides by z. z ²⁰²⁵ = z • z̅ = |z|² z is defined as z = a + b i , where a and b are each real numbers. ( a + b i )²⁰²⁵ = a² + b² Take both sides to the 1/2025 power. a + b i = ( a² + b² ) ¹ᐟ²⁰²⁵ On the right is a pure real number with 0 imaginary part. Therefore b = 0. a = ( a² ) ¹ᐟ²⁰²⁵ Raise to the power 2025. a²⁰²⁵ = a² Subtract a². a²⁰²⁵ - a² = 0 Factor. a² ( a²⁰²³ - 1 ) = 0 Use the zero product property. First factor: a = 0 Second factor: a²⁰²³ - 1 = 0 a²⁰²³ = 1 a = 1 , since it is a real number. answer z ∈ { 0, 1 }