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

Verified step by step guidance

1. Recall that the imaginary unit \(i\) has the property \(i^2 = -1\), and powers of \(i\) cycle every 4 steps: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the pattern repeats.
2. Rewrite the expression \(\frac{1}{i^{-11}}\) as \(i^{11}\) because dividing by \(i^{-11}\) is the same as multiplying by \(i^{11}\) (using the property \(\frac{1}{a^{-n}} = a^n\)).
3. To simplify \(i^{11}\), find the remainder when 11 is divided by 4, since powers of \(i\) repeat every 4: calculate \$11 \mod 4$.
4. Use the remainder from the previous step to rewrite \(i^{11}\) as one of \(i^0\), \(i^1\), \(i^2\), or \(i^3\), which correspond to \$1\(, \)i\(, \)-1\(, or \)-i$ respectively.
5. Express the simplified form of \(i^{11}\) based on the cycle and write the final simplified expression.

Key Concepts

Imaginary Unit and Powers of i

The imaginary unit i is defined as the square root of -1, with the property i² = -1. Powers of i cycle every four exponents: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1, then the pattern repeats. Understanding this cycle helps simplify expressions involving 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 allows rewriting expressions like i⁻¹ as 1/i, which is essential for simplifying powers with negative exponents.

Simplifying Complex Fractions

Simplifying expressions involving fractions with powers, such as 1/i⁻¹¹, requires applying exponent rules and properties of i. This often involves rewriting negative exponents, using the cyclic nature of i's powers, and reducing the expression to a standard form without imaginary units in the denominator.