Question

the fraction is (5^-3)^6 times 5^17 / 10^-2.

Answer 1

To simplify the expression \((5^{-3})^6 \times 5^{17} / 10^{-2}\), let's break it down step by step.

1. Simplify \((5^{-3})^6\):
\[
(5^{-3})^6 = 5^{-18}
\]

2. Now substitute this back into the expression:
\[
5^{-18} \times 5^{17} / 10^{-2}
\]

3. Use the property of exponents that states \(a^m \times a^n = a^{m+n}\):
\[
5^{-18} \times 5^{17} = 5^{-18 + 17} = 5^{-1}
\]

4. Now our expression looks like this:
\[
\frac{5^{-1}}{10^{-2}}
\]

5. Since \(10^{-2} = (2 \times 5)^{-2} = 2^{-2} \times 5^{-2}\), we have:
\[
10^{-2} = \frac{1}{10^2} = \frac{1}{100}
\]

6. Thus, we can rewrite the fraction:
\[
\frac{5^{-1}}{10^{-2}} = 5^{-1} \times 10^2 = 5^{-1} \times 100
\]

7. Now express \(5^{-1}\) as \(\frac{1}{5}\):
\[
\frac{1}{5} \times 100 = \frac{100}{5} = 20
\]

So, the simplified result of the given expression \((5^{-3})^6 \times 5^{17} / 10^{-2}\) is:
\[
\boxed{20}
\]