To simplify the expression \((5^{-3})^6 \cdot 5^{1710^{-2}}\) and express it with only positive exponents, we will apply the properties of exponents step by step.
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Simplify \((5^{-3})^6\): \[ (5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18} \]
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Simplify \(5^{1710^{-2}}\): First, we interpret \(1710^{-2}\). \[ 1710^{-2} = \frac{1}{1710^2} \] Thus, \[ 5^{1710^{-2}} = 5^{\frac{1}{1710^2}} \text{ (We will leave it as is for now).} \]
Now we have: \[ 5^{-18} \cdot 5^{1710^{-2}} \]
- Combine the two expressions using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\): \[ 5^{-18 + \frac{1}{1710^2}} \]
Now, we want to express this with only positive exponents. We can express \(5^{-18}\) as: \[ \frac{1}{5^{18}} \]
Thus, we rewrite the entire expression: \[ 5^{-18 + \frac{1}{1710^2}} = \frac{1}{5^{18 - \frac{1}{1710^2}}} \]
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Final expression with only positive exponents: \[ \frac{1}{5^{18 - \frac{1}{1710^2}}} \]
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Solving the expression: To find the exact numerical value, we can estimate \(1710^{-2}\): \[ 1710^2 = 2924100 \implies \frac{1}{1710^2} \approx 0 \]
Thus, \[ \text{For practical calculations: } 18 - \frac{1}{1710^2} \approx 18 \]
Hence, we get: \[ \frac{1}{5^{18}} \]
This is our final answer.
Summary
The equivalent expression with only positive exponents is: \[ \frac{1}{5^{18 - \frac{1}{1710^2}}} \] And numerically simplified (as much as needed) it approximates to: \[ \frac{1}{5^{18}} \]