determine the value of p such that the following equation is correct (3³)⁵ = 3π

The given equation is:

\[
(3^3)^5 = 3^p
\]

We need to determine the value of \( p \) such that this equation holds.

### Step-by-Step Solution:

1. Simplify the left-hand side:
   \[
   (3^3)^5 = 3^{3 \times 5} = 3^{15}
   \]

So, the equation becomes:

\[
3^{15} = 3^p
\]

2. Since the bases are the same, we can equate the exponents:

\[
15 = p
\]

Thus, the value of \( p \) is \( \boxed{15} \).

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Would you like further clarification or details on any step? 

Here are 5 related questions to expand your understanding:

1. How would you solve \( (2^4)^6 = 2^p \)?
2. Can you express \( (a^m)^n \) in a simplified form?
3. What is the general rule for solving \( (b^x)^y = b^z \)?
4. How do you apply logarithms to solve \( 5^x = 25 \)?
5. What would the value of \( p \) be if the equation was \( (4^2)^p = 4^{16} \)?

**Tip**: Always check for common bases when solving exponential equations—it allows you to easily compare the exponents!

Learn how to determine the value of p in the equation (3^3)^5 = 3^p with a detailed step-by-step solution. This problem involves exponentiation rules and solving exponential equations. Suitable for high school students.

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