Let's go through each question one by one:

1. Identify the correct statement based on the properties of exponents.

i) $2a^m \cdot a^n = 2a^{m+n}$
ii) $(2a)^m = 2^m a^m$
iii) $\frac{1}{2a^{-m}} = 2a^m$
iv) $\sqrt[m]{\frac{a^n}{b^n}} = \frac{a^{n/m}}{b^{n/m}}$

Correct answer: ii is true based on the property $(ab)^m = a^m \cdot b^m$.

2. Identify the correct statement based on the properties of exponents.

i) $a^0 = a = a^n$ for $a \neq 0$
ii) $\frac{a^m}{a^n} = a^{m-n}$
iii) $\frac{a^{-m}}{a^{-n}} = a^{n-m}$
iv) $\sqrt[b]{\frac{a^m}{b^m}} = \frac{a^{m/b}}{b^{m/b}}$
v) $\frac{a^m}{a^{-n}} = a^{m+n}$

Correct answer: ii is true based on the property $\frac{a^m}{a^n} = a^{m-n}$.

3. Simplify $10x^2 \cdot 2x^4 \cdot x^2$.

$10 \cdot 2 \cdot x^{2+4+2} = 20x^8$

Correct answer: A. 20x^8

4. Simplify $\frac{3}{6} x^5 \cdot 2x^3 \cdot x^{-4}$.

$\frac{3}{6} \cdot 2 \cdot x^{5+3-4} = x^4$

Correct answer: D. x^4

5. Simplify $\frac{1}{3} x^3 \cdot 2x \cdot \frac{2}{4} x^2$.

$\frac{1}{3} \cdot 2 \cdot \frac{2}{4} \cdot x^{3+1+2} = \frac{1}{3} \cdot 2 \cdot \frac{2}{4} x^6 = \frac{2}{6} x^6 = \frac{1}{3} x^6$

Correct answer: A. $\frac{2}{4} x^6$

Would you like further details on any of these solutions, or have any questions? Here are 5 related questions you might find interesting:

1. How do you simplify expressions with negative exponents?
2. What are the properties of exponents for division and multiplication?
3. How do you handle fractional exponents?
4. How do you combine terms with different bases?
5. What are the common mistakes when working with exponents?

Tip: Always remember to add or subtract exponents when multiplying or dividing like bases, respectively.