Let's break down and solve the problems from the worksheet step by step.

---

Part 1: Evaluating Exponential Expressions

a) $3^{-4}$

Solution:

Using the rule $a^{-n} = \frac{1}{a^n}$: $3^{-4} = \frac{1}{3^4} = \frac{1}{81}$

b) $5^{-2}$

Solution:

Similarly, applying $a^{-n} = \frac{1}{a^n}$: $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

c) $4^{-2} \times 2^3$

Solution:

$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ $2^3 = 8$ Multiplying them together: $\frac{1}{16} \times 8 = \frac{8}{16} = \frac{1}{2}$

d) $-25^{-3}$

Solution:

Applying the rule for negative exponents: $-25^{-3} = -\frac{1}{25^3} = -\frac{1}{15625}$

---

Part 2: Simplifying Expressions

a) $\frac{a^8 \times a^{-12}}{a^7}$

Solution:

Use the rule $a^m \times a^n = a^{m+n}$ for multiplication and $\frac{a^m}{a^n} = a^{m-n}$ for division. $a^8 \times a^{-12} = a^{8 + (-12)} = a^{-4}$ Now divide by $a^7$: $\frac{a^{-4}}{a^7} = a^{-4-7} = a^{-11}$ Final answer: $a^{-11}$

b) $\frac{2 x^{-2} 4 y^{-3}}{8 x^{12} y^{32}}$

Solution:

1. Simplify coefficients: $\frac{2 \times 4}{8} = 1$.
2. Simplify exponents for $x$ using $x^m / x^n = x^{m-n}$: $x^{-2 - 12} = x^{-14}$
3. Simplify exponents for $y$: $y^{-3 - 32} = y^{-35}$ Final answer: $x^{-14} y^{-35}$

c) $\frac{81 a^{-4} b^8}{143 a^2 b^{-33}}$

Solution:

1. Coefficients remain: $\frac{81}{143}$.
2. Simplify exponents for $a$: $a^{-4 - 2} = a^{-6}$
3. Simplify exponents for $b$: $b^{8 - (-33)} = b^{8 + 33} = b^{41}$ Final answer: [ \frac{81}{143} a^{-6