https://file.aiquickdraw.com/mathbot/file/1726116595887jmg3ym3t.jpg

The image contains a mathematical equation involving complex numbers. It asks to calculate the value of the following expression:

\[
\frac{i}{1 - 2i} = \boxed{\text{ケコ}} + \boxed{\text{サシ}} i
\]

Let's work through this expression step by step.

### Step 1: Multiply by the complex conjugate
To simplify \(\frac{i}{1 - 2i}\), we multiply both the numerator and the denominator by the complex conjugate of the denominator, which is \(1 + 2i\).

\[
\frac{i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{i(1 + 2i)}{(1 - 2i)(1 + 2i)}
\]

### Step 2: Simplify the denominator
The denominator becomes a difference of squares:

\[
(1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - (-4) = 1 + 4 = 5
\]

So the denominator is 5.

### Step 3: Expand the numerator
Now, expand the numerator:

\[
i(1 + 2i) = i \cdot 1 + i \cdot 2i = i - 2
\]

Thus, the numerator is \(i - 2\).

### Step 4: Final expression
Now, the expression becomes:

\[
\frac{i - 2}{5} = \frac{-2}{5} + \frac{i}{5}
\]

### Step 5: Compare with the original form
The expression can be written as:

\[
\frac{-2}{5} + \frac{1}{5}i
\]

So, comparing this with the form \(\boxed{\text{ケコ}} + \boxed{\text{サシ}} i\):

- \(\boxed{\text{ケコ}} = -\frac{2}{5}\)
- \(\boxed{\text{サシ}} = \frac{1}{5}\)

Thus, the values of the boxes are \(-\frac{2}{5}\) for "ケコ" and \(\frac{1}{5}\) for "サシ".

Would you like further details on any step?

### Related Questions:
1. How do you find the complex conjugate of a number?
2. What is the significance of multiplying by the conjugate in complex numbers?
3. Can you explain why the denominator becomes a difference of squares?
4. How does expanding the numerator work with imaginary units?
5. Why is the result written in the form \(\boxed{\text{ケコ}} + \boxed{\text{サシ}}i\)?

### Tip:
When simplifying complex fractions, always consider multiplying by the conjugate to remove imaginary numbers from the denominator.

Calculate the expression: \( \frac{i}{1 - 2i} = \boxed{\text{ケコ}} + \boxed{\text{サシ}} i \).

Learn how to simplify the complex expression \( \frac{i}{1 - 2i} \) by multiplying by the complex conjugate and breaking it down step-by-step. This problem covers key mathematical concepts like complex numbers, simplification, and algebraic manipulation. Suitable for high school students (Grades 10-12).

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