Simplify various integer powers of the imaginary unit i by using its cyclic properties.

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Simplifying Powers of the Imaginary Unit

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Simplify: (a) \(i^0, i^3, i^4\) (b) \(i^5, i^6, i^7, i^8\) (c) \(i^{39}, i^{174}, i^{252}, i^{317}\)

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What you need first

Basic arithmetic and division with remainders
Laws of exponents

What this problem tests

Tests the understanding of the cyclic nature of the powers of the imaginary unit and the application of the division algorithm to exponents.

This is a fundamental skill introduced when first learning about complex numbers, often serving as a prerequisite for linear algebra over the complex field.

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