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.