https://file.aiquickdraw.com/mathbot/file/1726074336459i6rlibk0.jpg

The expression in the image is \((3^4)^5\).

To simplify this, we apply the power of a power rule in exponents, which states:

\[
(a^m)^n = a^{m \cdot n}
\]

Thus, \((3^4)^5\) becomes:

\[
3^{4 \cdot 5} = 3^{20}
\]

Now, calculating \(3^{20}\):

\[
3^{20} = 3 \times 3 \times 3 \times \ldots \ (\text{20 times}) = 3,486,784,401
\]

So, \((3^4)^5 = 3^{20} = 3,486,784,401\).

Would you like more details or have any questions?

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Here are five related questions to deepen your understanding:

1. How do you simplify \((a^m)^n\) if both \(a\), \(m\), and \(n\) are variables?
2. Can you apply the power rule to expressions like \((2^3)^4\)? What is the result?
3. How does the power of a product rule differ from the power of a power rule?
4. What happens if the base in the expression is a fraction, such as \(\left(\frac{2}{3}\right)^4\)?
5. How do you solve problems involving negative exponents?

**Tip:** When working with large exponents, breaking down the expression using exponent rules helps manage calculations efficiently.

Learn how to simplify the exponential expression (3^4)^5 using the power of a power rule. This problem involves key exponentiation concepts and demonstrates step-by-step calculations for a clearer understanding. Suitable for students in Grades 8-10.

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