1/i(i-1) = 1/(i-1)-1/i . δειξε μου τα βήματα

To solve the equation:

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

We can proceed step by step as follows:

### Step 1: Simplify the left-hand side (LHS)

We begin with the left-hand side:

\[
\frac{1}{i(i-1)}
\]

This is already simplified.

### Step 2: Simplify the right-hand side (RHS)

Now, let's simplify the right-hand side:

\[
\frac{1}{i-1} - \frac{1}{i}
\]

We need to combine these two fractions by finding a common denominator. The common denominator between \(i - 1\) and \(i\) is \(i(i - 1)\). So, let's rewrite both fractions with the common denominator:

\[
\frac{1}{i-1} = \frac{i}{i(i-1)}, \quad \frac{1}{i} = \frac{i-1}{i(i-1)}
\]

Now, substitute these into the expression:

\[
\frac{i}{i(i-1)} - \frac{i-1}{i(i-1)}
\]

### Step 3: Combine the terms on the right-hand side

Now that we have a common denominator, we can combine the terms in the numerator:

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

### Step 4: Compare both sides

Now, we see that both sides are equal:

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

Thus, the equation is verified to be correct.

Would you like further clarification on any part?

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Here are 5 related questions to expand on this topic:

1. What are some examples of common denominators in fractions?
2. How can you solve an equation involving complex numbers like \(i\)?
3. What is the significance of \(i\) in mathematics (i.e., imaginary numbers)?
4. How do you combine fractions with different denominators?
5. What are some typical applications of imaginary numbers?

**Tip:** When combining fractions, always remember to find the least common denominator (LCD) to simplify your calculations!

1/i(i-1) = 1/(i-1) - 1/i . δειξε μου τα βήματα

Explore the detailed solution to the equation 1/i(i-1) = 1/(i-1) - 1/i, including steps on simplifying fractions and manipulating expressions with imaginary numbers. This problem is suitable for high school students in Grades 9-12.

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