To simplify the expression \(\left(3^{-8}\cdot 7^{3}\right)^{-2}\), we start by applying the property of exponents that states \((a \cdot b)^{-n} = a^{-n} \cdot b^{-n}\).
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Apply the exponent to each factor inside the parentheses: \[ \left(3^{-8}\cdot 7^{3}\right)^{-2} = \left(3^{-8}\right)^{-2} \cdot \left(7^{3}\right)^{-2} \]
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Simplify each term by applying the power of a power property, which states \((a^m)^n = a^{m \cdot n}\): \[ \left(3^{-8}\right)^{-2} = 3^{-8 \cdot -2} = 3^{16} \] \[ \left(7^{3}\right)^{-2} = 7^{3 \cdot -2} = 7^{-6} \]
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Now combine the results from the above steps: \[ 3^{16} \cdot 7^{-6} \]
Now we check which option matches our result:
- (Choice A) \(21^{10}\)
- (Choice B) \(3^{16}\cdot 7^{-6}\)
- (Choice C) \(\dfrac{7^6}{3^{16}}\)
The equivalent expression is: \[ \boxed{B} \quad 3^{16} \cdot 7^{-6} \]