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### **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:
\[
-25^{-3} = -\frac{1}{25^3} = -\frac{1}{15625}
\]

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### **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

Please solve the whole worksheet in detail step by step

Explore detailed solutions for simplifying and evaluating exponents and radical expressions. This worksheet covers essential algebra concepts suitable for Grades 9-10 students, including laws of exponents and algebraic simplification.

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