Question

Apply the properties of integer exponents to generate an equivalent expression to (5^-3)^6 x 5^17/10^-2 with only positive exponents, then solve the expression.

Answer 1

To simplify the expression \((5^{-3})^6 \cdot 5^{17} / 10^{-2}\), we will apply the properties of integer exponents step by step.

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

2. **Rewrite the expression**: Replace \((5^{-3})^6\) in the original expression.
\[
5^{-18} \cdot 5^{17} / 10^{-2}
\]

3. **Use the product of powers property**: The property states that \(a^m \cdot a^n = a^{m+n}\).
\[
5^{-18} \cdot 5^{17} = 5^{-18 + 17} = 5^{-1}
\]

So the expression now becomes:
\[
\frac{5^{-1}}{10^{-2}}
\]

4. **Rewrite the division as multiplication by the reciprocal**: We can rewrite the expression using the property \(\frac{a^m}{b^n} = a^m \cdot b^{-n}\).
\[
5^{-1} \cdot 10^{2}
\]

5. **Convert \(10^{2}\)**: We can express \(10\) as \(2 \cdot 5\):
\[
10^2 = (2 \cdot 5)^2 = 2^2 \cdot 5^2 = 4 \cdot 25 = 100
\]

But since we only need it in terms of bases, we retain \(10^2\).

6. **Combine the exponents**: We know that \(10^2 = 100\).
\[
5^{-1} \cdot 10^{2} = \frac{10^2}{5}
\]

Now we can express \(5^{-1}\) as \(\frac{1}{5}\).
\[
\frac{10^2}{5} = \frac{100}{5} = 20
\]

Thus, the equivalent expression with positive exponents simplifies to:
\[
\boxed{20}
\]