Let's solve each of the given expressions one by one.

a. $2x^2 \cdot 7x^4 \cdot (3x)^2$

1. Simplify each term:

   • $2x^2$ stays the same.
   • $7x^4$ stays the same.
   • $(3x)^2 = 9x^2$

2. Combine all the terms: $2x^2 \cdot 7x^4 \cdot 9x^2 = (2 \cdot 7 \cdot 9) \cdot (x^2 \cdot x^4 \cdot x^2)$ $= 126x^{2+4+2}$ $= 126x^8$

b. $\left(\frac{2p}{q}\right)^2 \cdot \left(q^4\right)^{\frac{1}{2}} \cdot p^2$

1. Simplify each term:

   • $\left(\frac{2p}{q}\right)^2 = \frac{4p^2}{q^2}$
   • $\left(q^4\right)^{\frac{1}{2}} = q^2$
   • $p^2$ stays the same

2. Combine all the terms: $\frac{4p^2}{q^2} \cdot q^2 \cdot p^2 = 4p^2 \cdot p^2$ $= 4p^{2+2}$ $= 4p^4$

c. $y^2 \cdot (xy^3)^{\frac{1}{3}} \cdot \left(\frac{1}{x}\right)^2$

1. Simplify each term:

   • $y^2$ stays the same.
   • $(xy^3)^{\frac{1}{3}} = x^{\frac{1}{3}}y$
   • $\left(\frac{1}{x}\right)^2 = \frac{1}{x^2} = x^{-2}$

2. Combine all the terms: $y^2 \cdot x^{\frac{1}{3}}y \cdot x^{-2} = x^{\frac{1}{3}} x^{-2} y^2 y$ $= x^{\frac{1}{3} - 2} y^{2+1}$ $= x^{-\frac{5}{3}} y^3$

d. $(abc)^2 \cdot \frac{3}{a^2b^{\frac{1}{2}}c^{\frac{1}{2}}} \cdot \frac{b^{\frac{5}{2}}}{2a^2}$

1. Simplify each term:

   • $(abc)^2 = a^2b^2c^2$
   • $\frac{3}{a^2b^{\frac{1}{2}}c^{\frac{1}{2}}}$ stays the same
   • $\frac{b^{\frac{5}{2}}}{2a^2}$ stays the same

2. Combine all the terms: $a^2b^2c^2 \cdot \frac{3}{a^2b^{\frac{1}{2}}c^{\frac{1}{2}}} \cdot \frac{b^{\frac{5}{2}}}{2a^2}$

   First, combine the fractions: $\frac{3}{a^2b^{\frac{1}{2}}c^{\frac{1}{2}}} \cdot \frac{b^{\frac{5}{2}}}{2a^2} = \frac{3b^{\frac{5}{2}}}{2a^4b^{\frac{1}{2}}c^{\frac{1}{2}}}$ $= \frac{3b^{\frac{5}{2}-\frac{1}{2}}}{2a^4c^{\frac{1}{2}}} = \frac{3b^2}{2a^4c^{\frac{1}{2}}}$

   Now, combine this with $a^2b^2c^2$: $a^2b^2c^2 \cdot \frac{3b^2}{2a^4c^{\frac{1}{2}}} = \frac{3a^{2-4}b^{2+2}c^{2-\frac{1}{2}}}{2} = \frac{3a^{-2}b^4c^{\frac{3}{2}}}{2}$ $= \frac{3b^4c^{\frac{3}{2}}}{2a^2}$

Summary

• a: $126x^8$
• b: $4p^4$
• c: $x^{-\frac{5}{3}} y^3$
• d: $\frac{3b^4c^{\frac{3}{2}}}{2a^2}$

Would you like any further details or have any questions?

Here are some related questions you might find helpful:

1. How do you simplify expressions involving multiple variables and exponents?
2. What are the rules for multiplying powers with the same base?
3. How do you handle fractional exponents in algebraic expressions?
4. What are the steps to simplify complex fractions?
5. How do you apply the distributive property to exponents?
6. What are the common mistakes to avoid when simplifying expressions?
7. How do you combine terms with different bases and exponents?
8. Can you explain how to simplify expressions involving negative exponents?

Tip: Always break down complex expressions into smaller parts and simplify step by step to avoid mistakes.