(3a 
3
 b 
5
 c) 
2
 (4ab 
2
 c 
3
 )(2ab 
−5
 ) 
0

The expression you've written appears to involve powers of terms with variables. Let me break it down for you:

\[
(3a^3 b^5 c)^2 \times (4ab^2 c^3) \times (2ab^{-5})^0
\]

### Step-by-step breakdown:

1. **First term: \( (3a^3 b^5 c)^2 \)**
   - Apply the power of 2 to each factor inside the parentheses:
     \[
     (3a^3 b^5 c)^2 = 3^2 a^{3 \times 2} b^{5 \times 2} c^2 = 9a^6 b^{10} c^2
     \]

2. **Second term: \( (4ab^2 c^3) \)**
   - This is just \( 4ab^2 c^3 \), no changes are necessary.

3. **Third term: \( (2ab^{-5})^0 \)**
   - Any expression raised to the power of 0 is 1, so this becomes:
     \[
     (2ab^{-5})^0 = 1
     \]

Now, combine the simplified terms:

\[
9a^6 b^{10} c^2 \times 4ab^2 c^3 \times 1
\]

### Simplify further:

- Combine the constants: \( 9 \times 4 = 36 \)
- For \( a \): \( a^6 \times a = a^{6+1} = a^7 \)
- For \( b \): \( b^{10} \times b^2 = b^{10+2} = b^{12} \)
- For \( c \): \( c^2 \times c^3 = c^{2+3} = c^5 \)

Thus, the fully simplified expression is:

\[
36a^7 b^{12} c^5
\]

---

Would you like a more detailed breakdown of any of the steps or need further help? Here are some related questions:

1. How do you handle negative exponents in expressions?
2. What happens if there’s a fractional exponent in an expression?
3. How do you simplify expressions with multiple terms raised to the same power?
4. How do you distribute exponents over multiplication or division in an expression?
5. What are the basic exponent rules that help with simplifying these types of problems?

**Tip**: Always remember to apply exponents to each factor inside parentheses individually when dealing with powers.

Learn how to simplify the expression (3a^3 b^5 c)^2 (4ab^2 c^3) (2ab^{-5})^0 using exponent laws and algebraic rules. This step-by-step guide covers key concepts such as power of a product, zero exponents, and combining like terms. Suitable for students in Grades 9-11.

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