1. **Problem statement:** Simplify and evaluate numerical expressions involving positive and negative integer indices. 2. **Formula and rules:** For any nonzero number $a$ and integers $m$ and $n$: - $a^m \times a^n = a^{m+n}$ (product rule) - $\left(a^m\right)^n = a^{m \times n}$ (power of a power) - $a^{-n} = \frac{1}{a^n}$ (negative exponent rule) - $a^0 = 1$ (zero exponent rule) 3. **Example:** Simplify and evaluate $2^{-3} \times 2^4$. 4. **Step-by-step solution:** - Using the product rule: $$2^{-3} \times 2^4 = 2^{-3+4} = 2^1$$ - Simplify exponent: $$2^1 = 2$$ 5. **Another example:** Evaluate $\left(3^2\right)^{-1}$. 6. **Step-by-step solution:** - Using power of a power rule: $$\left(3^2\right)^{-1} = 3^{2 \times (-1)} = 3^{-2}$$ - Using negative exponent rule: $$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$ 7. **Summary:** When simplifying expressions with integer indices, apply the exponent rules carefully, especially for negative exponents which represent reciprocals. **Final answers:** - $2^{-3} \times 2^4 = 2$ - $\left(3^2\right)^{-1} = \frac{1}{9}$