Simplify each power of i. 1/i^-11

Verified step by step guidance

1. Recall that the imaginary unit \(i\) satisfies the property \(i^4 = 1\), which means powers of \(i\) repeat every 4 steps.
2. Rewrite the expression \(\frac{1}{i^{-11}}\) as \(i^{11}\) by using the property \(\frac{1}{i^a} = i^{-a}\), so \(\frac{1}{i^{-11}} = i^{11}\).
3. Simplify the exponent 11 by reducing it modulo 4, since powers of \(i\) cycle every 4: calculate \$11 \mod 4$.
4. Express \(i^{11}\) as \(i^{(4 \times 2) + 3} = (i^4)^2 \times i^3\), and since \(i^4 = 1\), this simplifies to \(i^3\).
5. Recall that \(i^3 = i^2 \times i = (-1) \times i = -i\), so the simplified form of the original expression is \(-i\).

Key Concepts

Properties of the Imaginary Unit i

The imaginary unit i is defined as the square root of -1, with the property that i² = -1. Powers of i cycle every four exponents: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1, then the pattern repeats. Understanding this cyclical behavior is essential for simplifying powers of i.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a⁻ⁿ = 1/aⁿ. Applying this rule helps rewrite expressions with negative powers into more manageable forms for simplification.

Simplifying Complex Fractions

Simplifying expressions like 1 divided by a power involves rewriting the denominator using exponent rules and then applying properties of exponents and imaginary numbers. This process often requires converting negative exponents and using the cyclical nature of i to reduce the expression to a standard form.