$$  $$$$ \left(13^2\right)^5\text{ }\cdot13^{-11} $$$$  $$$$ \cdot $$simplified

To simplify the expression $\left(13^2\right)^5 \cdot 13^{-11}$:

1. Apply the power of a power property $\left(a^m\right)^n = a^{m \cdot n}$:

   \[
   \left(13^2\right)^5 = 13^{2 \cdot 5} = 13^{10}
   \]

2. Now, the expression is:

   \[
   13^{10} \cdot 13^{-11}
   \]

3. Apply the product of powers property $a^m \cdot a^n = a^{m+n}$:

   \[
   13^{10} \cdot 13^{-11} = 13^{10 - 11} = 13^{-1}
   \]

4. Finally, simplify:

   \[
   13^{-1} = \frac{1}{13}
   \]

So, the simplified expression is $\frac{1}{13}$.

Would you like further details, or do you have any questions?

Here are five related questions you might find helpful:
1. How do you simplify expressions involving exponents?
2. What are the rules for multiplying and dividing exponents with the same base?
3. How do you convert a negative exponent into a fraction?
4. How can you apply exponent rules in more complex algebraic expressions?
5. What is the significance of zero exponents in simplifying expressions?

**Tip:** Always remember that a negative exponent indicates the reciprocal of the base raised to the positive exponent.

Learn how to simplify the expression (13^2)^5 * 13^-11 step-by-step with detailed explanations. This problem involves applying exponent rules such as the power of a power property and the product of powers property. Suitable for Grades 7-9 students learning algebraic expressions.

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