Evaluate : i⁻⁵⁰

To evaluate \( i^{-50} \), we first need to understand the properties of the imaginary unit \( i \), which is defined as \( i = \sqrt{-1} \).

Understanding Powers of i

The powers of \( i \) cycle through four values:

• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1

Finding the Equivalent Positive Power

Since the powers of \( i \) repeat every four, we can find \( i^{-50} \) by first converting the negative exponent:

Using the property \( i^{-n} = \frac{1}{i^n} \), we have:

i^-50 = 1 / i⁵⁰

Calculating i⁵⁰

Next, we need to find \( i^{50} \). We can do this by finding the remainder of 50 when divided by 4:

50 mod 4 = 2

This means:

i⁵⁰ = i² = -1

Final Calculation

Now we substitute back into our expression:

i^-50 = 1 / (-1) = -1

Thus, the value of \( i^{-50} \) is -1.