Decide whether each statement is true or false. If false, correct the right side of the equation. i¹² = 1

Verified step by step guidance

1. Recall that the imaginary unit \(i\) is defined such that \(i^2 = -1\).
2. To simplify \(i^{12}\), express the exponent 12 in terms of multiples of 4 because powers of \(i\) cycle every 4: \(i^4 = 1\).
3. Write \(i^{12}\) as \((i^4)^3\) since \(12 = 4 \times 3\).
4. Use the fact that \((i^4)^3 = 1^3\) because \(i^4 = 1\).
5. Therefore, \(i^{12} = 1^3 = 1\), so the statement \(i^{12} = 1\) is true.

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 steps: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1, then the pattern repeats. Understanding this cycle is essential to simplify higher powers of i.

Modular Arithmetic for Exponentiation

When dealing with powers of i, exponents can be reduced modulo 4 because the powers repeat every 4 steps. For example, i^12 can be simplified by calculating 12 mod 4, which equals 0, so i^12 = i^0 = 1. This technique simplifies evaluating large exponents.

Evaluating and Verifying Equations

To determine if an equation like i^12 = 1 is true, substitute the simplified value of i^12 using the power cycle and modular arithmetic. If the left and right sides match, the statement is true; otherwise, correct the right side accordingly. This process ensures accurate verification.