Simplify the expression (p5r2)4(−7p3r4)(6pr3)(𝑝5𝑟2)4(-7𝑝3𝑟4)(6𝑝𝑟3) . Write the variables in alphabetical order.

1. Break Down the Problem

We need to simplify the expression $(p^5r^2)^4(-7p^3r^4)(6pr^3)$. First, we will expand each part of the expression and then combine like terms.

2. Relevant Concepts

• Apply the power rule: $(a^m)^n = a^{mn}$
• Combine coefficients and like bases in multiplication.

3. Analysis and Detail

1. Expand $(p^5r^2)^4$: $(p^5r^2)^4 = p^{5 \cdot 4} r^{2 \cdot 4} = p^{20} r^8$

2. Rewrite the expression with parts expanded: $p^{20}r^8 \cdot (-7p^3r^4) \cdot (6pr^3)$

3. Combine coefficients: $-7 \cdot 6 = -42$

4. Combine the powers of $p$: $p^{20} \cdot p^3 \cdot p^1 = p^{20 + 3 + 1} = p^{24}$

5. Combine the powers of $r$: $r^8 \cdot r^4 \cdot r^3 = r^{8 + 4 + 3} = r^{15}$

4. Verify and Summarize

Now, we combine the results to get the simplified expression: $-42p^{24}r^{15}$

Final Answer

The simplified expression is $-42p^{24}r^{15}$.