To simplify the expression \( (6\sqrt{2})^4 \) using the power rule, you first apply the power to both the coefficient and the radical.
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Power of the coefficient: \[ 6^4 = 1296 \]
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Power of the square root: \[ (\sqrt{2})^4 = (\sqrt{2})^2 \times (\sqrt{2})^2 = 2 \times 2 = 4 \] Alternatively, you can directly use the property that \( (\sqrt{a})^n = a^{n/2} \): \[ (\sqrt{2})^4 = 2^{4/2} = 2^2 = 4 \]
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Combine the results: \[ (6\sqrt{2})^4 = 6^4 \times (\sqrt{2})^4 = 1296 \times 4 \] Calculate: \[ 1296 \times 4 = 5184 \]
Thus, the simplified form of \( (6\sqrt{2})^4 \) is \( 5184 \).