How to Simplify i^94: Step-by-Step Guide for Students
Quick Answer
To simplify i^94, recognize that i is the imaginary unit where i^2 = -1. By rewriting i^94 as (i^2)^47, you find that it equals (-1)^47, which simplifies to -1.
To simplify the expression i^94, it's important to understand the properties of the imaginary unit 'i'. The imaginary unit 'i' is defined as the square root of -1, and one of its key properties is that i^2 equals -1. This fundamental property helps us work with higher powers of 'i'.
First, let's rewrite i^94 in a more manageable form. Notice that 94 is an even number, and we can express i^94 as (i^2)^47. This step is crucial because it allows us to utilize the property of 'i' efficiently.
Now, substituting in the known value of i^2, we have:
(i^2)^47 = (-1)^47.
Next, we need to evaluate (-1)^47. Since 47 is an odd number, when you raise -1 to an odd exponent, the result is always -1. This is because multiplying -1 an odd number of times will yield a negative result. Thus, we find that:
(-1)^47 = -1.
So, after simplifying, we arrive at the conclusion that i^94 = -1.
### Why It Matters
Understanding how to simplify powers of 'i' can be very useful, especially in higher mathematics, physics, and engineering. Complex numbers play a significant role in various fields, including electrical engineering and quantum physics. By mastering these concepts, you gain a better grasp of more advanced topics involving complex numbers.
### Additional Example
For further practice, letโs simplify another power of 'i'. Consider i^50.
1. Recognize that 50 is even, so we can express it as (i^2)^25.
2. Substitute in the value of i^2: (i^2)^25 = (-1)^25.
3. Since 25 is odd, (-1)^25 = -1.
4. Therefore, i^50 = -1.
By following this process, you can simplify any power of 'i' efficiently. Remember, the key is recognizing the even or odd nature of the exponent and applying the property of i^2 = -1.
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