What is the value of the expression [tex]$i^0 \times i^1 \times i^2 \times i^3 \times i^4$[/tex]?
Answer (1)
Simplify the expression using exponent rules: i 0 × i 1 × i 2 × i 3 × i 4 = i 0 + 1 + 2 + 3 + 4 = i 10 .
Use the property i 4 = 1 to simplify i 10 = i 8 × i 2 = ( i 4 ) 2 × i 2 = 1 2 × i 2 = i 2 .
Recall that i 2 = − 1 .
Therefore, the final answer is − 1 .
Explanation
Understanding the Problem We are asked to find the value of the expression i 0 × i 1 × i 2 × i 3 × i 4 , where i is the imaginary unit, defined as i = − 1 .
Properties of i Recall the properties of i : i 0 = 1
i 1 = i
i 2 = − 1
i 3 = − i
i 4 = 1
Simplifying the Expression We can simplify the expression using the properties of exponents: i 0 × i 1 × i 2 × i 3 × i 4 = i 0 + 1 + 2 + 3 + 4 = i 10
Simplifying the Exponent Now, we need to find the value of i 10 . We know that i 4 = 1 , so we can write i 10 as: i 10 = i 8 + 2 = i 8 × i 2
Calculating the Final Value Since i 4 = 1 , we have i 8 = ( i 4 ) 2 = 1 2 = 1 . Therefore, i 10 = i 8 × i 2 = 1 × i 2 = i 2 = − 1
Final Answer Thus, the value of the expression i 0 × i 1 × i 2 × i 3 × i 4 is − 1 .
Examples
Understanding imaginary numbers is crucial in electrical engineering, especially when analyzing alternating current (AC) circuits. In AC circuits, voltage and current oscillate, and imaginary numbers help represent the phase difference between them. For example, when calculating impedance in an AC circuit, you often encounter expressions involving complex numbers, where the imaginary part accounts for the reactance due to capacitors and inductors. By using imaginary numbers, engineers can simplify the analysis and design of complex electrical systems, ensuring efficient and stable operation.
