To simplify the expression \((5^{-3})^6 \cdot \frac{5^{17}}{10^{-2}}\) and generate an equivalent expression with only positive exponents, we will follow these steps:
- Apply the power of a power property of exponents, which states that \((a^m)^n = a^{m \cdot n}\).
- Combine the exponents using the multiplication and division of the exponents using the properties \(a^m \cdot a^n = a^{m+n}\) and \(\frac{a^m}{a^n} = a^{m-n}\).
- Replace any negative exponents with positive exponents.
Step 1: Simplify \((5^{-3})^6\)
Using the power of a power property:
\[ (5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18} \]
So, we can rewrite the expression as:
\[ 5^{-18} \cdot \frac{5^{17}}{10^{-2}} \]
Step 2: Simplify the division with \(10^{-2}\)
Next, we can rewrite the division:
\[ \frac{5^{17}}{10^{-2}} = 5^{17} \cdot 10^{2} \]
Thus, we can now write the expression as:
\[ 5^{-18} \cdot 5^{17} \cdot 10^{2} \]
Step 3: Combine the powers of 5
Now we will combine the powers of \(5\):
\[ 5^{-18} \cdot 5^{17} = 5^{-18 + 17} = 5^{-1} \]
Now we can express this with positive exponents:
\[ 5^{-1} = \frac{1}{5} \]
Step 4: Rewrite the entire expression
Putting it all together, we have:
\[ \frac{1}{5} \cdot 10^{2} \]
Since \(10^{2} = 100\), we can multiply:
\[ \frac{1}{5} \cdot 100 = \frac{100}{5} = 20 \]
Thus, the final answer to the original expression with only positive exponents is:
\[ \boxed{20} \]
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