To determine whether the equation \(5^{-5}\cdot (\sqrt [3]{5})^{12}=\frac {1}{5}\) is true, we will simplify the left-hand side of the equation step by step.
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First, we start with the term \((\sqrt[3]{5})^{12}\). We can rewrite this using the property of exponents: \[ \sqrt[3]{5} = 5^{1/3} \] Therefore, \[ (\sqrt[3]{5})^{12} = (5^{1/3})^{12} = 5^{(1/3) \cdot 12} = 5^{12/3} = 5^4. \]
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Now we substitute this back into the left-hand side of our equation: \[ 5^{-5} \cdot (\sqrt[3]{5})^{12} = 5^{-5} \cdot 5^4. \]
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We can simplify this product using the property of exponents \(a^m \cdot a^n = a^{m+n}\): \[ 5^{-5} \cdot 5^4 = 5^{-5 + 4} = 5^{-1}. \]
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Now we have \(5^{-1}\) on the left-hand side. We know that: \[ 5^{-1} = \frac{1}{5}. \]
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This shows that the left-hand side simplifies to \(\frac{1}{5}\), which is equal to the right-hand side.
Thus, the equation is true, and we conclude with: \[ \boxed{1} \]