what is the value of the expression $i^0 \\times i^1 \\times i^2 \\times i^3 \\times i^4$?\n○ 1\n○ -1\n○…

Explanation:

Step1: Recall properties of imaginary unit (i)

We know that (i^0 = 1) (any non - zero number to the power of 0 is 1), (i^1=i), (i^2=- 1), (i^3=i^2\times i=-i), (i^4=(i^2)^2 = (-1)^2 = 1).

Step2: Simplify the exponents of (i) in the product

First, use the property of exponents (a^m\times a^n=a^{m + n}). So (i^0\times i^1\times i^2\times i^3\times i^4=i^{0 + 1+2 + 3+4}=i^{10}).

Step3: Reduce (i^{10}) using the cycle of (i)

The powers of (i) repeat every 4: (i^1 = i), (i^2=-1), (i^3=-i), (i^4 = 1), (i^5=i^4\times i = i), and so on. We can write (i^{10}=i^{4\times2 + 2}). Since (i^{4k}=(i^4)^k = 1^k=1) for integer (k), then (i^{4\times2+2}=i^{8 + 2}=i^8\times i^2=(i^4)^2\times i^2). Substituting (i^4 = 1) and (i^2=-1), we get ((1)^2\times(-1)=-1). We can also calculate the sum of exponents first: (0 + 1+2+3 + 4=10), then (i^{10}=(i^2)^5=(-1)^5=-1).

Answer:

(-1) (corresponding to the option with value (-1))