Question: Evaluate the numerical value of the expression without using a calculator: $$(-4)^{-2} \times 2^{5}$$
1. **State the problem:** We need to evaluate the expression $$(-4)^{-2} \times 2^{5}$$ without using a calculator.
2. **Recall the rules:**
- A negative exponent means taking the reciprocal: $$a^{-n} = \frac{1}{a^{n}}$$.
- Powers of numbers can be simplified by multiplying the bases if they are the same.
3. **Rewrite the expression using the negative exponent rule:**
$$(-4)^{-2} = \frac{1}{(-4)^{2}}$$
4. **Calculate $$(-4)^2$$:**
$$(-4)^2 = (-4) \times (-4) = 16$$
5. **Substitute back:**
$$(-4)^{-2} = \frac{1}{16}$$
6. **Calculate $$2^5$$:**
$$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$$
7. **Multiply the two results:**
$$\frac{1}{16} \times 32 = \frac{32}{16}$$
8. **Simplify the fraction:**
$$\frac{32}{16} = \cancel{\frac{32}{16}} = 2$$
**Final answer:** $$2$$