https://file.aiquickdraw.com/mathbot/file/1726585182472anc8l8j3.jpg

The expression in the image is:

\[
A = \frac{(6^8)^2 \cdot (6^4)^5}{(6^{20})^4 \cdot (6^3)^7}
\]

### Step-by-step solution:

1. **Simplify the powers**:
   Use the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\).

   - \((6^8)^2 = 6^{8 \cdot 2} = 6^{16}\)
   - \((6^4)^5 = 6^{4 \cdot 5} = 6^{20}\)
   - \((6^{20})^4 = 6^{20 \cdot 4} = 6^{80}\)
   - \((6^3)^7 = 6^{3 \cdot 7} = 6^{21}\)

   So, the expression becomes:

   \[
   A = \frac{6^{16} \cdot 6^{20}}{6^{80} \cdot 6^{21}}
   \]

2. **Combine powers in the numerator and denominator**:
   Use the product of powers rule, which states \(a^m \cdot a^n = a^{m+n}\).

   - Numerator: \(6^{16} \cdot 6^{20} = 6^{16+20} = 6^{36}\)
   - Denominator: \(6^{80} \cdot 6^{21} = 6^{80+21} = 6^{101}\)

   Now the expression becomes:

   \[
   A = \frac{6^{36}}{6^{101}}
   \]

3. **Simplify the fraction**:
   Use the quotient of powers rule, which states \(\frac{a^m}{a^n} = a^{m-n}\).

   \[
   A = 6^{36 - 101} = 6^{-65}
   \]

Thus, the simplified value of \(A\) is:

\[
A = 6^{-65}
\]

This represents a very small number, as it is the reciprocal of \(6^{65}\).

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Do you need any further explanation or details on the solution?

Here are 5 related questions to expand on this:

1. How would the result change if the base number was not 6 but another value?
2. What is the significance of negative exponents in mathematics?
3. Can you express \(6^{-65}\) as a decimal approximation?
4. How does the power of powers rule differ from the product of powers rule?
5. What happens if the numerator and denominator were swapped in the original expression?

**Tip**: When simplifying powers, always look for opportunities to combine terms using exponent rules before performing any complex calculations!

Simplify the following expression: (6^8)^2 * (6^4)^5 / ((6^20)^4 * (6^3)^7)

This problem involves simplifying a complex powers expression using exponentiation rules, such as the power of a power, product of powers, and quotient of powers rules. A step-by-step breakdown is provided to simplify the given expression.

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