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.

(1 point)
(5^−3)^6⋅ 5^17/10^−2=

Answer 1

Let's simplify the expression \((5^{-3})^6 \cdot 5^{17} / 10^{-2}\) step-by-step, applying the properties of integer exponents.

1. Apply the power of a power property \((a^m)^n = a^{m \cdot n}\):
\[
(5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18}
\]

2. Combine the exponents of \(5\) using the property \(a^m \cdot a^n = a^{m+n}\):
\[
5^{-18} \cdot 5^{17} = 5^{-18 + 17} = 5^{-1}
\]

3. The expression is now:
\[
\frac{5^{-1}}{10^{-2}}
\]

4. Remember that \(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\) or you can use the property \(\frac{1}{a^{-n}} = a^{n}\) to rewrite the division as multiplication:
\[
5^{-1} \cdot 10^{2}
\]

5. Rewrite \(10^{2}\) as \( (2 \cdot 5)^{2} = 2^{2} \cdot 5^{2} = 4 \cdot 25 = 100\):
\[
5^{-1} \cdot 100
\]

6. Now simplify \(5^{-1} = \frac{1}{5}\):
\[
\frac{1}{5} \cdot 100 = \frac{100}{5} = 20
\]

Thus, the equivalent expression with only positive exponents simplifies to \(20\).

The final answer is:
\[
\boxed{20}
\]