how did you reduce to the power of 2^30 to 2^15 in step 6?

It looks like there was a misunderstanding in the simplification process. The power \(2^{30}\) should not be reduced incorrectly. Let's correct that step-by-step simplification again, paying attention to powers:

1. Simplify inside the parentheses:
   \[
   (2^2 \times 3^3)^5 \rightarrow (4 \times 27)^5 \rightarrow (108)^5
   \]

2. Expand other terms:
   \[
   4^7 \rightarrow (2^2)^7 \rightarrow 2^{14}
   \]
   \[
   8 \rightarrow 2^3
   \]
   \[
   25^3 \rightarrow (5^2)^3 \rightarrow 5^6
   \]
   \[
   (8 \times 25^3)^2 \rightarrow (2^3 \times 5^6)^2 \rightarrow 2^6 \times 5^{12}
   \]

3. Combine all terms:
   \[
   (108)^5 \times 2^{14} \times 5^3 \times 2^6 \times 5^{12}
   \]

4. Simplify the bases:
   - First, note that \( 108 = 2^2 \times 3^3 \) 
   - So, \( (108)^5 = (2^2 \times 3^3)^5 \rightarrow 2^{10} \times 3^{15} \)

   Now we have:
   \[
   2^{10} \times 3^{15} \times 2^{14} \times 5^3 \times 2^6 \times 5^{12}
   \]

5. Combine like terms correctly:
   \[
   2^{10 + 14 + 6} \times 3^{15} \times 5^{3 + 12} \rightarrow 2^{30} \times 3^{15} \times 5^{15}
   \]

6. Rewrite in a single form:
   \[
   (2 \times 3 \times 5)^{15} \rightarrow (30)^{15}
   \]

So the correct simplification, \(2^{30}\), should indeed stay as \(30^{15}\) and not \(2^{15}\).

Therefore, the corrected simplified expression is indeed \( 30^{15} \).

Do you want details or have any questions? Here are five questions you might ask next:
1. How do you simplify expressions with multiple bases?
2. Can you explain the laws of exponents in more detail?
3. Can you solve another exponent simplification problem?
4. What is the process for simplifying complex expressions?
5. How can we verify the correctness of each step in simplification?

**Tip:** Always verify each step carefully when combining exponents to ensure accuracy in the final result.

Explore the correct simplification process from 2^30 to 30^15 step-by-step. This explanation covers the laws of exponents and how to simplify complex exponential expressions. Suitable for high school students and those preparing for advanced mathematics.

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