To simplify the expression \((5^{-3})^6 \times 5^{17} / 10^{-2}\), let's break it down step by step.
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Simplify \((5^{-3})^6\): \[ (5^{-3})^6 = 5^{-18} \]
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Now substitute this back into the expression: \[ 5^{-18} \times 5^{17} / 10^{-2} \]
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Use the property of exponents that states \(a^m \times a^n = a^{m+n}\): \[ 5^{-18} \times 5^{17} = 5^{-18 + 17} = 5^{-1} \]
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Now our expression looks like this: \[ \frac{5^{-1}}{10^{-2}} \]
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Since \(10^{-2} = (2 \times 5)^{-2} = 2^{-2} \times 5^{-2}\), we have: \[ 10^{-2} = \frac{1}{10^2} = \frac{1}{100} \]
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Thus, we can rewrite the fraction: \[ \frac{5^{-1}}{10^{-2}} = 5^{-1} \times 10^2 = 5^{-1} \times 100 \]
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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: \[ \boxed{20} \]
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