To simplify the expression \((5 \sqrt{2})(3 \sqrt{6})\), we can first multiply the numerical coefficients (5 and 3) and then the square roots (\(\sqrt{2}\) and \(\sqrt{6}\)).
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Multiply the coefficients: \[ 5 \times 3 = 15 \]
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Multiply the square roots: \[ \sqrt{2} \times \sqrt{6} = \sqrt{2 \times 6} = \sqrt{12} \]
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Combine the two results: \[ (5 \sqrt{2})(3 \sqrt{6}) = 15 \sqrt{12} \]
Now, we can simplify \(\sqrt{12}\): \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Now, substitute this back into our expression: \[ 15 \sqrt{12} = 15 \times 2 \sqrt{3} = 30 \sqrt{3} \]
Therefore, the equivalent expression is: \[ 30 \sqrt{3} \]
Looking at the answer options, the correct choice is: D. \(30 \sqrt{3}\)
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