To simplify the expression \( \sqrt{40} - 3\sqrt{3} + \sqrt{290} \), let's simplify each square root and combine them where possible.
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Simplify \( \sqrt{40} \): \[ \sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10} \]
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Simplify \( \sqrt{290} \): \[ \sqrt{290} = \sqrt{29 \cdot 10} = \sqrt{29} \cdot \sqrt{10} = \sqrt{290} \] (Note that \( 29 \) is prime, so it doesn't simplify further.)
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Putting it all together: \[ 2\sqrt{10} - 3\sqrt{3} + \sqrt{290} \]
Since \( \sqrt{290} \) cannot be simplified and does not combine with the other terms, we keep it in its current form.
Thus, the expression is: \[ 2\sqrt{10} + \sqrt{290} - 3\sqrt{3} \]
Among the given responses, the one that corresponds closest to our simplification is not directly listed, so let's analyze options based on what we can extract:
- \( 810−−√−3–√ \) is equal to \( 8\sqrt{10} - \sqrt{3} \). This does not match our expression.
- \( -410−−√−3–√ \) is equal to \( -4\sqrt{10} - \sqrt{3} \). This does not match our expression.
- \( 2127−−−√ \) suggests \( 2\sqrt{127} \), which again does not align.
- \( 1310−−√−3–√ \) corresponds to \( \sqrt{13}\sqrt{10}-\sqrt{3} \), which does not correlate to our terms.
Based on this analysis, it seems there is no exact match with the options provided, but the steps to simplify the original expression have all been covered.
Thus, for a direct answer, the simplified expression is: \[ 2\sqrt{10} + \sqrt{290} - 3\sqrt{3} \]
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