Let's simplify the expression step by step:

The expression you provided is:

$(5a^2b^3c^4)^4(6a^3b^4c^2)$

Step 1: Distribute the exponent in the first part $(5a^2b^3c^4)^4$

Apply the exponent to each factor inside the parentheses:

$(5a^2b^3c^4)^4 = 5^4a^{2 \cdot 4}b^{3 \cdot 4}c^{4 \cdot 4}$ This simplifies to: $5^4a^8b^{12}c^{16}$

Step 2: Multiply with the second part $(6a^3b^4c^2)$

Now, multiply the result with $6a^3b^4c^2$:

$5^4a^8b^{12}c^{16} \cdot 6a^3b^4c^2$

Multiply the constants and combine like terms for $a$, $b$, and $c$:

• $5^4 \cdot 6 = 625 \cdot 6 = 3750$
• For $a$, combine $a^8 \cdot a^3 = a^{8+3} = a^{11}$
• For $b$, combine $b^{12} \cdot b^4 = b^{12+4} = b^{16}$
• For $c$, combine $c^{16} \cdot c^2 = c^{16+2} = c^{18}$

Thus, the simplified expression is:

$3750a^{11}b^{16}c^{18}$

Final Answer:

$3750a^{11}b^{16}c^{18}$

Would you like any further details on this process?

Follow-up questions:

1. How would the expression change if the powers of $a$, $b$, and $c$ were negative instead of positive?
2. Can you simplify expressions involving more than two terms with exponents?
3. How do you handle variables when exponents are fractions or decimals?
4. What is the rule for multiplying terms with exponents that have different bases?
5. How do you deal with terms that have no common bases or exponents?

Tip: Always distribute exponents carefully, especially when multiplying terms with variables. It helps to avoid errors when combining like terms!