The maximum number of real roots of the equation `x^(2n) -1 = 0`, is

Options

A. 2
B. 3
C. n
D. 2n

Answer

To solve the equation \( x^{2n} - 1 = 0 \) and find the maximum number of real roots, we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^{2n} - 1 = 0 \]
This can be rewritten as:
\[ x^{2n} = 1 \]

### Step 2: Identify the roots
The equation \( x^{2n} = 1 \) implies that \( x \) can take values that satisfy this equation. The solutions to \( x^{2n} = 1 \) are the \( 2n \)-th roots of unity.

### Step 3: Determine the values of \( x \)
The general solutions for \( x^{2n} = 1 \) can be expressed as:
\[ x = 1 \quad \text{and} \quad x = -1 \]
However, since we are looking for real roots, we need to consider the nature of the roots.

### Step 4: Analyze the number of real roots
The equation \( x^{2n} = 1 \) has two distinct real roots:
1. \( x = 1 \)
2. \( x = -1 \)

### Step 5: Conclusion
Thus, the maximum number of real roots for the equation \( x^{2n} - 1 = 0 \) is:
\[ \text{Maximum number of real roots} = 2 \]

### Final Answer
The maximum number of real roots of the equation \( x^{2n} - 1 = 0 \) is **2**.

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