What is the value of the expression [tex]$i^0 \times i^1 \times i^2 \times i^3 \times i^4$[/tex]?
Answer (2)
The value of the expression i 0 × i 1 × i 2 × i 3 × i 4 simplifies to − 1 through the use of properties of exponents and the cyclical nature of powers of the imaginary unit i .
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Rewrite the expression using properties of exponents: i 0 × i 1 × i 2 × i 3 × i 4 = i 0 + 1 + 2 + 3 + 4 .
Calculate the sum of the exponents: 0 + 1 + 2 + 3 + 4 = 10 .
Simplify the expression: i 10 = i 4 × 2 + 2 = i 2 .
Since i 2 = − 1 , 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 = \tIsqrt − 1 . We know that the powers of i cycle through four values: i 0 = 1 , i 1 = i , i 2 = − 1 , i 3 = − i , and i 4 = 1 .
Simplifying the Expression Using the properties of exponents, we can rewrite the expression as i 0 + 1 + 2 + 3 + 4 .
Calculating the Sum of Exponents Now, we need to calculate the sum of the exponents: 0 + 1 + 2 + 3 + 4 = 10 .
Finding the Remainder So, the expression simplifies to i 10 . Since the powers of i cycle every four powers, we need to find the remainder when 10 is divided by 4. We have 10 = 4 × 2 + 2 , so the remainder is 2.
Final Calculation Therefore, i 10 = i 2 . Since i 2 = − 1 , the value of the expression is -1.
Examples
Understanding imaginary numbers and their powers 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, the impedance of a circuit, which is the opposition to the flow of current, can be expressed using complex numbers involving the imaginary unit 'i'. By calculating powers of 'i', engineers can simplify complex circuit equations and design efficient power systems.
