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Apply the Properties of Integer Exponents to generate an equivalent expression to (5^-3)^6 times 5^17/10^-2 with only positive exponents, then solve the expression. Show your work.

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To solve the expression \((5^{-3})^6 \times \frac{5^{17}}{10^{-2}}\) using the properties of integer exponents, we will follow these steps:

  1. Apply the exponent rule \((a^m)^n = a^{mn}\) to simplify \((5^{-3})^6\).
  2. Simplify the fraction \(\frac{5^{17}}{10^{-2}}\).
  3. Combine the results and simplify further.

Step 1: Simplify \((5^{-3})^6\)

Using the exponent rule \((a^m)^n = a^{mn}\):

\[ (5^{-3})^6 = 5^{-18} \]

Step 2: Simplify \(\frac{5^{17}}{10^{-2}}\)

Recall that \(10^{n} = 2^{n} \cdot 5^{n}\). Therefore, we can rewrite \(10^{-2}\) as:

\[ 10^{-2} = \frac{1}{10^2} = \frac{1}{2^2 \cdot 5^2} = \frac{1}{4 \cdot 25} = \frac{1}{100} \]

Hence,

\[ \frac{5^{17}}{10^{-2}} = 5^{17} \cdot 10^2 = 5^{17} \cdot 100 = 5^{17} \cdot (10^2) = 5^{17} \cdot (2^2 \cdot 5^2) = 5^{17} \cdot 2^2 \cdot 5^2 \]

This further simplifies to:

\[ 5^{17} \cdot 4 \cdot 5^2 = 4 \cdot 5^{17 + 2} = 4 \cdot 5^{19} \]

Step 3: Combine Results

Now we can combine \(5^{-18}\) and \(4 \cdot 5^{19}\):

\[ 5^{-18} \cdot (4 \cdot 5^{19}) = 4 \cdot 5^{-18 + 19} = 4 \cdot 5^{1} \]

Step 4: Rewrite with Positive Exponents

Since \(5^1 = 5\), we have:

\[ 4 \cdot 5 = 20 \]

Thus, the final equivalent expression with only positive exponents is:

\[ \boxed{20} \]

the / means fraction

Thank you for clarifying. Let's resolve the expression correctly, accounting for the division properly. The expression we need to simplify is:

\[ (5^{-3})^6 \times \frac{5^{17}}{10^{-2}} \]

Step 1: Simplify \((5^{-3})^6\)

Using the exponent rule \((a^m)^n = a^{mn}\):

\[ (5^{-3})^6 = 5^{-18} \]

Step 2: Rewrite \(\frac{5^{17}}{10^{-2}}\)

We can rewrite \(10^{-2}\) using its prime factorization:

\[ 10 = 2 \times 5 \quad \Rightarrow \quad 10^{-2} = (2 \times 5)^{-2} = 2^{-2} \cdot 5^{-2} \]

Thus, we can rewrite the fraction:

\[ \frac{5^{17}}{10^{-2}} = 5^{17} \cdot 10^2 = 5^{17} \cdot (2^2 \cdot 5^2) \]

This results in:

\[ \frac{5^{17}}{10^{-2}} = 5^{17} \cdot 2^2 \cdot 5^2 = 4 \cdot 5^{17 + 2} = 4 \cdot 5^{19} \]

Step 3: Combine the Results

Now we can combine \(5^{-18}\) and \(4 \cdot 5^{19}\):

\[ 5^{-18} \cdot (4 \cdot 5^{19}) = 4 \cdot 5^{-18 + 19} = 4 \cdot 5^1 \]

Step 4: Final Calculation

Thus, we have:

\[ 4 \cdot 5 = 20 \]

Therefore, the final simplified expression is:

\[ \boxed{20} \]