Question

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

Show your work.

Answer 1

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:

1. Apply the power of a power property of exponents, which states that \((a^m)^n = a^{m \cdot n}\).
2. Combine the exponents using the multiplication and division of the exponents using the properties \(a^m \cdot a^n = a^{m+n}\) and \(\frac{a^m}{a^n} = a^{m-n}\).
3. 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:

\[
\boxed{20}
\]