9) \( \sqrt{\square \sqrt{490}}=\sqrt[166]{490} \)

To solve the equation, first simplify the left side. We have \( \sqrt{\square \sqrt{490}} \). We know that \( \sqrt{490} \) can be expressed as \( \sqrt{49 \cdot 10} = 7 \sqrt{10} \). Therefore, if we let \( x = \sqrt{490} \), the left side becomes \( \sqrt{a \cdot x} \), where \( x = \sqrt{490} \). Next, let's take a look at the right-hand side: \( \sqrt[166]{490} \). This is the same as \( 490^{1/166} \). To solve for \( \square \), we equate both sides: \[ \sqrt{\square \cdot \sqrt{490}} = \sqrt[166]{490} \] Now, let's square both sides to eliminate the square root: \[ \square \cdot \sqrt{490} = \left(490^{1/166}\right)^2 \] Which simplifies to: \[ \square \cdot \sqrt{490} = 490^{2/166} = 490^{1/83} \] Now, isolate \( \square \): \[ \square = \frac{490^{1/83}}{\sqrt{490}} \] Since \( \sqrt{490} = 490^{1/2} \), we have: \[ \square = 490^{1/83 - 1/2} \] To combine the exponents, convert \( 1/2 \) into a fraction with a denominator of 83: \[ \square = 490^{1/83 - 41.5/83} = 490^{-40.5/83} = 490^{-40.5/83} \] Thus, the expression actually asks for \( \square \) representing a value derived from \( 490^{-40.5/83} \). The answer will retain that form—completeness and context really matter in understanding the uses of square roots in both theoretical and practical applications!