Let's solve the questions step by step:

Question 18:

Expression: $(2 \times \frac{1}{2} \times 4)^n$
Find $n$ such that the expression equals 1.

1. Simplify the base:
   $2 \times \frac{1}{2} \times 4 = 1 \times 4 = 4$.

   So, the expression becomes $4^n$.

2. For $4^n = 1$, the only possible value of $n$ is $n = 0$ because any number raised to the power of 0 is 1.

Answer: $b. 0$

Question 19:

Expression: $4^0 \cdot 5^{-2} \cdot 6^{-1} \cdot 100^{\frac{1}{2} - 1}$

1. Simplify each term:

   • $4^0 = 1$,
   • $5^{-2} = \frac{1}{25}$,
   • $6^{-1} = \frac{1}{6}$,
   • $100^{\frac{1}{2} - 1} = 100^{-\frac{1}{2}} = \frac{1}{\sqrt{100}} = \frac{1}{10}$.

2. Multiply all terms:
   $1 \cdot \frac{1}{25} \cdot \frac{1}{6} \cdot \frac{1}{10} = \frac{1}{25 \cdot 6 \cdot 10} = \frac{1}{1500}.$

Answer: $d. \frac{1}{1500}$

Question 20:

Expression: $(5x^3y^2)^0$

1. Any expression raised to the power 0 is 1 (as long as the base is not 0).

Answer: $b. 1$

Would you like detailed explanations for any of these steps or similar problems?

Here are 5 related questions to explore:

1. What is the result of simplifying $(3^2 \cdot 4^3)^{-1}$?
2. How do you simplify an expression like $a^{-2}b^0c^{\frac{1}{2}}$?
3. If $2^x = 8$, find $x$.
4. Solve $(x^2)^3 \cdot x^{-4}$ for $x$.
5. What does $(5^{-1} \cdot 10^0)$ simplify to?

Tip: Always simplify terms inside brackets or parentheses before applying exponents.