Problem 177

Analysis

This problem tests the multiplication and division operations of complex numbers in algebraic form and examines the basic concepts of complex numbers. It is a fundamental question.

We will directly utilize the multiplication and division operations of complex numbers in algebraic form to simplify the complex number $\frac{2+i}{1-2i}$ and obtain the answer.

Step-by-Step Solution

1. Multiply the numerator and the denominator by the conjugate of the denominator, which is $(1+2i)$. This helps to eliminate the imaginary part in the denominator:

$$\frac{2+i}{1-2i} = \frac{(2+i)(1+2i)}{(1-2i)(1+2i)}$$

2. Apply the distributive law (also known as FOIL method) to both the numerator and the denominator:

$$\frac{2+i}{1-2i} = \frac{2 \cdot 1 + 2 \cdot 2i + i \cdot 1 + i \cdot 2i}{1 \cdot 1 - 1 \cdot 2i + (-2i) \cdot 1 + (-2i) \cdot 2i}$$

3. Simplify the expression:

$$\frac{2+i}{1-2i} = \frac{2 + 4i + i - 2}{1 - 2i + 2i - 4i^2}$$

4. Recall that $i^2 = -1$, and apply this to the expression:

$$\frac{2+i}{1-2i} = \frac{2 + 5i - 2}{1 + 4}$$

5. Combine like terms and simplify further:

$$\frac{2+i}{1-2i} = \frac{5i}{5}$$

6. Cancel out the common factor of $5$ from the numerator and the denominator:

$$\frac{2+i}{1-2i} = i$$

Thus, the answer is: $\boxed{i}$