Math Problem Statement

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

Solution

To simplify the expression (132)51311\left(13^2\right)^5 \cdot 13^{-11}:

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

    (132)5=1325=1310\left(13^2\right)^5 = 13^{2 \cdot 5} = 13^{10}

  2. Now, the expression is:

    1310131113^{10} \cdot 13^{-11}

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

    13101311=131011=13113^{10} \cdot 13^{-11} = 13^{10 - 11} = 13^{-1}

  4. Finally, simplify:

    131=11313^{-1} = \frac{1}{13}

So, the simplified expression is 113\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.

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Math Problem Analysis

Mathematical Concepts

Exponents

Formulas

Power of a power property
Product of powers property

Theorems

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Suitable Grade Level

Grades 7-9