Rationalize and Simplify the Given Expressions:
Solutions:
1. Use the product rule
\[\sqrt{2}\sqrt{6} + 3 \sqrt{12} = \sqrt{2.6} + 3 \sqrt{12}\]
\[\sqrt{12} + 3 \sqrt{12} = 12\sqrt{1 + 3} = 4 \sqrt{4. 3}\]
\[4\sqrt{4} \sqrt{3} = 4 . 2 \sqrt{3} = 8 \sqrt{3}\]
2. Write 14 and 63 as products of prime numbers 14 = 2 x 7 ; 63 = 3^2 x 7 and substitute
\[\frac{3 \sqrt{14} + 4 \sqrt{63}}{3 \sqrt{7}} = \frac{3 \sqrt{2 . 7} + 4 \sqrt{3^2 . 7}}{3 \sqrt{7}} \]
\[\frac{3 \sqrt{2} \sqrt{7} + 4 . 3 \sqrt{7}}{3 \sqrt{7}} \]
\[\frac{ \sqrt{7}(3 \sqrt{2} + 12)}{3 \sqrt{7}} \]
\[\frac{ \sqrt{7}(3 \sqrt{2}}{3 \sqrt{7}} + \frac{ \sqrt{7}( 12)}{3 \sqrt{7}} \]
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