Let's simplify the expression \(-8x^{-3} + 12x^{-6} + 1\).

### Step-by-step simplification:

1. The expression contains negative exponents, which we can rewrite as fractions:
   \[
   -8x^{-3} = \frac{-8}{x^3}, \quad 12x^{-6} = \frac{12}{x^6}
   \]
   So the expression becomes:
   \[
   \frac{-8}{x^3} + \frac{12}{x^6} + 1
   \]

2. This is the simplified version of the given expression. If you're looking to combine these terms further, you'd need a common denominator.

### Finding a common denominator:

- The common denominator between \(x^3\) and \(x^6\) is \(x^6\).
- Rewrite each term with \(x^6\) as the denominator:
   \[
   \frac{-8}{x^3} = \frac{-8x^3}{x^6}, \quad \frac{12}{x^6} = \frac{12}{x^6}
   \]
   So the expression becomes:
   \[
   \frac{-8x^3}{x^6} + \frac{12}{x^6} + 1
   \]

3. Combine the terms:
   \[
   \frac{-8x^3 + 12}{x^6} + 1
   \]

Thus, the simplified form of the expression is:
\[
\frac{-8x^3 + 12}{x^6} + 1
\]

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

1. What is the general rule for negative exponents?
2. How do you find a common denominator for fractional expressions?
3. Can we factor the numerator \(-8x^3 + 12\) further?
4. What are the properties of exponents for multiplication and division?
5. How would this expression behave as \(x\) approaches 0?

**Tip:** When simplifying expressions with negative exponents, always try to rewrite them as fractions first to make the operations clearer!

Discover how to simplify the expression -8x^-3 + 12x^-6 + 1. This problem involves algebraic concepts, such as negative exponents and fractional expressions. Follow a step-by-step solution to understand the properties of exponents and how to find a common denominator for fractional expressions. Suitable for students in Grades 10-12.

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