Simplify each power of i. i^-13

Verified step by step guidance

1. Recall that the powers of the imaginary unit $i$ cycle every four: $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$.
2. To simplify $i^{-13}$, first convert the negative exponent to a positive one by using the property $i^{-n}=\frac{1}{i^n}$.
3. Calculate $i^{13}$ by finding the remainder of 13 divided by 4, since the powers of $i$ repeat every 4.
4. Perform the division: 13 divided by 4 gives a quotient of 3 and a remainder of 1.
5. Thus, $i^{13}=i^1=i$, and $i^{-13}=\frac{1}{i}=-i$ (since $\frac{1}{i}=-i$ by multiplying numerator and denominator by $i$).

Key Concepts

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -1. It is a fundamental concept in complex numbers, allowing for the extension of real numbers to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0. Powers of 'i' cycle through four values: i^0 = 1, i^1 = i, i^2 = -1, and i^3 = -i.

Properties of Exponents

Properties of exponents govern how to manipulate powers of numbers. Key rules include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and negative exponents (a^-n = 1/a^n). Understanding these properties is essential for simplifying expressions involving exponents, including negative powers.

Cyclic Nature of Powers of i

The powers of 'i' exhibit a cyclic pattern every four terms. Specifically, i^n can be simplified by finding the remainder of n when divided by 4. This means that for any integer n, i^n can be expressed as one of the four values: 1, i, -1, or -i, depending on whether the remainder is 0, 1, 2, or 3, respectively. This property is crucial for simplifying expressions like i^-13.