To simplify the expression \((7^{-3})^7 \times 7^{20} / 6^{-2}\) and express it with positive exponents, we'll use the properties of integer exponents.
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Apply the power of a power rule: \((a^m)^n = a^{m \cdot n}\). \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]
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Combine the terms with the same base: We now have \(7^{-21} \times 7^{20}\). Using the property \(a^m \times a^n = a^{m+n}\): \[ 7^{-21} \times 7^{20} = 7^{-21 + 20} = 7^{-1} \]
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Convert the exponent to positive: Remember that \(a^{-n} = \frac{1}{a^n}\): \[ 7^{-1} = \frac{1}{7} \]
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Simplify the denominator: Now we simplify \(6^{-2}\) as well: \[ 6^{-2} = \frac{1}{6^2} = \frac{1}{36} \]
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Combine everything into a single expression: \[ \frac{1}{7} \div \frac{1}{36} \]
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Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{1}{7} \times 36 = \frac{36}{7} \]
Thus, the expression \((7^{-3})^7 \times 7^{20} / 6^{-2}\) simplifies and evaluates to: \[ \frac{36}{7} \]
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