To simplify the expression \( 2(R - 1)(R^3 - 1) = (R^2 - 1)^2 \), let's go through the steps methodically.

### Step 1: Factor the expressions
First, let's factor the terms on both sides of the equation.

1. **Factoring \( R^3 - 1 \):**
   \( R^3 - 1 \) is a difference of cubes, so it can be factored as:
   \[
   R^3 - 1 = (R - 1)(R^2 + R + 1)
   \]

2. **Factoring \( R^2 - 1 \):**
   \( R^2 - 1 \) is a difference of squares, so it can be factored as:
   \[
   R^2 - 1 = (R - 1)(R + 1)
   \]
   Therefore, \( (R^2 - 1)^2 \) becomes:
   \[
   (R^2 - 1)^2 = [(R - 1)(R + 1)]^2 = (R - 1)^2(R + 1)^2
   \]

### Step 2: Substitute the factored forms into the equation
Now, substitute the factored forms into the original equation:
\[
2(R - 1)(R - 1)(R^2 + R + 1) = (R - 1)^2(R + 1)^2
\]

### Step 3: Simplify both sides
1. On the left-hand side, combine the \( (R - 1) \) terms:
   \[
   2(R - 1)^2(R^2 + R + 1)
   \]

2. The right-hand side remains as:
   \[
   (R - 1)^2(R + 1)^2
   \]

Thus, the equation now looks like:
\[
2(R - 1)^2(R^2 + R + 1) = (R - 1)^2(R + 1)^2
\]

### Step 4: Divide both sides by \( (R - 1)^2 \)
As long as \( R \neq 1 \) (since dividing by zero is undefined), we can divide both sides of the equation by \( (R - 1)^2 \), giving:
\[
2(R^2 + R + 1) = (R + 1)^2
\]

### Step 5: Expand and simplify
1. Expand \( (R + 1)^2 \):
   \[
   (R + 1)^2 = R^2 + 2R + 1
   \]

2. So the equation becomes:
   \[
   2(R^2 + R + 1) = R^2 + 2R + 1
   \]

3. Expanding the left-hand side:
   \[
   2(R^2 + R + 1) = 2R^2 + 2R + 2
   \]

   Therefore, the equation is now:
   \[
   2R^2 + 2R + 2 = R^2 + 2R + 1
   \]

### Step 6: Move all terms to one side
Subtract \( R^2 + 2R + 1 \) from both sides:
\[
2R^2 + 2R + 2 - (R^2 + 2R + 1) = 0
\]

Simplifying:
\[
R^2 + 1 = 0
\]

### Step 7: Solve for \( R \)
The equation \( R^2 + 1 = 0 \) implies:
\[
R^2 = -1
\]
Thus, \( R = \pm i \), where \( i \) is the imaginary unit.

### Final Answer:
The simplified equation results in \( R = \pm i \).

Would you like further details on any of the steps? Here are some related questions:

1. How do you factor the difference of cubes and squares?
2. What is the significance of imaginary numbers in algebra?
3. Can you explain the conditions under which division by a term is allowed?
4. How does expanding binomials work in polynomial equations?
5. What are some real-world applications of imaginary numbers?

**Tip:** Always check for special conditions (like division by zero) when simplifying algebraic expressions.

Learn how to simplify the equation 2(R-1)(R³-1) = (R²-1)² through factoring techniques and solve for R. This problem covers advanced algebra concepts, including the difference of cubes and squares, and introduces imaginary numbers. Suitable for Grades 10-12.

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