Simplify the expression (2g^2n^2)(-7p^3r^4)^2(6pr^3)(5a^2b^3c^4)(6a^3b^4c^2).

Steps to Solve

1. Simplify each term Start with the individual terms:

• The first term is (2g^2n^2).
• The second term is ((-7p^3r^4)^2). Calculate this: $$ (-7)^2 \cdot (p^3)^2 \cdot (r^4)^2 = 49p^6r^8. $$

2. Calculate the third term The third term is (6pr^3), which remains as is.

3. Combine the fourth term The fourth term is (5a^2b^3c^4), which also remains unchanged.

4. Process the fifth term The fifth term is (6a^3b^4c^2), which stays the same.

5. Combine all terms together Now combine all simplified terms together: $$ (2g^2n^2) \cdot (49p^6r^8) \cdot (6pr^3) \cdot (5a^2b^3c^4) \cdot (6a^3b^4c^2). $$

6. Multiply the coefficients Multiply the numerical coefficients: $$ 2 \cdot 49 \cdot 6 \cdot 5 \cdot 6 = 2 \times 49 = 98,\ 98 \times 6 = 588,\ 588 \times 5 = 2940,\ 2940 \times 6 = 17640. $$

7. Combine like terms Now combine the variables:

• For (p): (p^6 \cdot p^1 = p^{7}),
• For (r): (r^8 \cdot r^3 = r^{11}),
• For (a): (a^2 \cdot a^3 = a^{5}),
• For (b): (b^3 \cdot b^4 = b^{7}),
• For (c): (c^4 \cdot c^2 = c^{6}).

8. Write the final expression Combine everything into the final expression: $$ 17640g^2n^2p^{7}r^{11}a^{5}b^{7}c^{6}. $$