Which of the following is an imaginary number? i^24; The simplest form of the imaginary number i^45; i^30; (1/i) ^199; i^26+i^28; (1/i) ^15; 5i^7+4i^-1.

Steps to Solve

1. Understanding the Powers of $i$

The imaginary unit $i$ is defined such that $i^2 = -1$. The powers of $i$ repeat every four terms:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$

We can reduce higher powers of $i$ to these values by taking the exponent modulo 4.

2. Calculating Each Expression

• (1) For $i^{24}$:

  Calculate $24 \mod 4 = 0 \Rightarrow i^{24} = i^0 = 1$.

• (2) For $i^{45}$:

  Calculate $45 \mod 4 = 1 \Rightarrow i^{45} = i^1 = i$.

• (3) For $i^{30}$:

  Calculate $30 \mod 4 = 2 \Rightarrow i^{30} = i^2 = -1$.

• (4) For $\left(\frac{1}{i}\right)^{199}$:

  Use $\frac{1}{i} = -i$, so $\left(-i\right)^{199}$. Calculate $199 \mod 4 = 3 \Rightarrow (-i)^{3} = -i^{3} = -(-i) = i$.

• (5) For $i^{26} + i^{28}$:

  Calculate $26 \mod 4 = 2 \Rightarrow i^{26} = -1$ and $28 \mod 4 = 0 \Rightarrow i^{28} = 1$. So, $-1 + 1 = 0$.

• (6) For $\left(\frac{1}{i}\right)^{15}$:

  Similar to step 4, $\left(-i\right)^{15}$: $15 \mod 4 = 3 \Rightarrow (-i)^{3} = -i^{3} = -(-i) = i$.

• (7) For $5i^7 + 4i^{-1}$:

  Calculate $7 \mod 4 = 3 \Rightarrow i^{7} = -i$ and $i^{-1} = -i$. Therefore, $5(-i) + 4(-i) = -5i - 4i = -9i$.

3. Compiling Results

After calculating, we summarize:

• (1) $1$
• (2) $i$
• (3) $-1$
• (4) $i$
• (5) $0$
• (6) $i$
• (7) $-9i$