Let's simplify the given expression step by step:

Expression:
$(3mn^2)^2 (2m^0n)^3$

Step 1: Simplify each term separately.

1. First term: $(3mn^2)^2$
   Use the property of exponents $(ab)^c = a^c \cdot b^c$:
   $(3mn^2)^2 = 3^2 \cdot m^2 \cdot (n^2)^2 = 9m^2n^4$

2. Second term: $(2m^0n)^3$

   • Simplify $m^0 = 1$ (anything raised to 0 is 1), so:
     $2m^0n = 2n$
   • Apply the exponent $3$:
     $(2n)^3 = 2^3 \cdot n^3 = 8n^3$

Step 2: Combine the simplified terms.

Now multiply the results:
$(3mn^2)^2 \cdot (2m^0n)^3 = (9m^2n^4) \cdot (8n^3)$

Step 3: Multiply coefficients and variables.

1. Multiply the coefficients: $9 \cdot 8 = 72$.
2. Combine $m^2$ (no $m$ terms from the second factor):
   $m^2$
3. Combine $n^4 \cdot n^3$ using the property $a^b \cdot a^c = a^{b+c}$:
   $n^{4+3} = n^7$

Final Result:

$72m^2n^7$

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Let me know if you'd like a detailed breakdown of any step! 😊

Related Questions:

1. What are the rules for simplifying exponents like $(a^m)^n$?
2. Can you explain why $m^0 = 1$ for any $m \neq 0$?
3. How do you multiply terms with different bases but the same exponent?
4. Can you simplify $(4x^3y)^2 (3xy^2)^3$?
5. How do you handle negative exponents in expressions?

Tip: Always simplify terms inside parentheses first before applying exponents!