https://file.aiquickdraw.com/mathbot/file/1725477269652vbjgsmog.jpg

The expression provided in the image is:

\[
\frac{3}{1 + x^n} + \frac{3}{1 + x^{-n}}
\]

Let's simplify this expression.

### Step 1: Common Denominator
To combine the two fractions, we need a common denominator. The common denominator is the product of the two individual denominators:

\[
\text{Common Denominator} = (1 + x^n)(1 + x^{-n})
\]

### Step 2: Multiply Numerators by Opposite Denominators
The expression becomes:

\[
\frac{3(1 + x^{-n}) + 3(1 + x^n)}{(1 + x^n)(1 + x^{-n})}
\]

### Step 3: Simplify the Numerator
Expand the numerator:

\[
3(1 + x^{-n}) + 3(1 + x^n) = 3 + 3x^{-n} + 3 + 3x^n = 6 + 3x^n + 3x^{-n}
\]

So, the expression simplifies to:

\[
\frac{6 + 3x^n + 3x^{-n}}{(1 + x^n)(1 + x^{-n})}
\]

### Step 4: Simplify the Denominator
Let's simplify the denominator:

\[
(1 + x^n)(1 + x^{-n}) = 1 + x^n + x^{-n} + 1 = 2 + x^n + x^{-n}
\]

### Step 5: Final Simplified Expression
Thus, the expression becomes:

\[
\frac{6 + 3x^n + 3x^{-n}}{2 + x^n + x^{-n}}
\]

We can factor out a 3 from the numerator:

\[
\frac{3(2 + x^n + x^{-n})}{2 + x^n + x^{-n}}
\]

Finally, since the numerator and denominator are the same, the expression simplifies to:

\[
\boxed{3}
\]

This is the simplified form of the given expression.

Would you like any more details or have any questions?

Here are some follow-up questions:
1. What happens if the exponents \( n \) are different in the two fractions?
2. How would this expression change if there was a subtraction instead of addition?
3. Can you explore what happens as \( x \) approaches 0 or infinity?
4. What if we changed the coefficients from 3 to another number?
5. How can you generalize this solution for any similar rational expressions?

**Tip:** When simplifying fractions, always look for common factors in the numerator and denominator that can cancel out to simplify the expression further.

Simplify the expression 3/(1 + x^n) + 3/(1 + x^-n)

Explore the step-by-step process to simplify the rational expression 3/(1 + x^n) + 3/(1 + x^-n). This problem covers key algebraic concepts, including adding fractions with different denominators and simplification of expressions involving exponents. Suitable for students in Grades 9-12.

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