Simplify each power of i. 1/i17

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 36

2장, 문제 36

Simplify each power of i. 1/i¹⁷

검증된 단계별 안내

Recall that the imaginary unit \(i\) has the property \(i^2 = -1\), and powers of \(i\) cycle every 4: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).

To simplify \(\frac{1}{i^{17}}\), first focus on simplifying the denominator \(i^{17}\) by reducing the exponent modulo 4, since powers of \(i\) repeat every 4.

Calculate \(17 \mod 4\) to find the equivalent smaller exponent: \(17 \div 4\) leaves a remainder of 1, so \(i^{17} = i^{1} = i\).

Rewrite the expression as \(\frac{1}{i}\), which can be simplified by multiplying numerator and denominator by the complex conjugate of the denominator or by using the fact that \(\frac{1}{i} = -i\).

Express the final simplified form by applying the multiplication and simplifying any resulting terms.

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Imaginary Unit i and its Powers

The imaginary unit i is defined as the square root of -1. Its powers cycle every four steps: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1, then the pattern repeats. Understanding this cyclicity helps simplify expressions involving powers of i.

Negative and Fractional Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent, so i⁻ⁿ = 1/iⁿ. Recognizing this allows rewriting expressions like 1/i¹⁷ as i⁻¹⁷, facilitating simplification using the power cycle of i.

Modular Arithmetic for Exponent Reduction

Since powers of i repeat every 4, exponents can be reduced modulo 4 to simplify calculations. For example, 17 mod 4 equals 1, so i¹⁷ = i¹. This technique simplifies large exponents by focusing on their remainder when divided by 4.