Evaluate: $\frac{3+\sqrt{2}}{3\sqrt{2}(\frac{1}{\sqrt{3}})}$
Understand the Problem
The question asks us to evaluate the mathematical expression shown in the image, which involves simplifying a fraction with radicals in both the numerator and denominator. We will simplify this in steps.
Answer
$$ \frac{3\sqrt{6}+2\sqrt{3}}{6} $$
Answer for screen readers
$$ \frac{3\sqrt{6}+2\sqrt{3}}{6} $$
Steps to Solve
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Rewrite the expression The expression is: $$ \frac{3+\sqrt{2}}{3\sqrt{2}(\frac{1}{\sqrt{3}})} $$
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Simplify the denominator Multiply the terms in the denominator: $$ 3\sqrt{2} \cdot \frac{1}{\sqrt{3}} = \frac{3\sqrt{2}}{\sqrt{3}} $$
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Rationalize the denominator Multiply the denominator and numerator by $\sqrt{3}$: $$ \frac{3\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{6}}{3} = \sqrt{6} $$
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Substitute simplified denominator back into the expression $$ \frac{3+\sqrt{2}}{\sqrt{6}} $$
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Rationalize the denominator (again) Multiply the numerator and denominator by $\sqrt{6}$: $$ \frac{3+\sqrt{2}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{3\sqrt{6}+\sqrt{12}}{6} $$
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Simplify the radical $\sqrt{12}$ can be simplified to $2\sqrt{3}$: $$ \frac{3\sqrt{6}+2\sqrt{3}}{6} $$
$$ \frac{3\sqrt{6}+2\sqrt{3}}{6} $$
More Information
The simplified form of the given expression is $\frac{3\sqrt{6}+2\sqrt{3}}{6}$. This can also be written as $\frac{\sqrt{3}}{6}(3\sqrt{2}+2)$ or $\frac{\sqrt{6}}{2} + \frac{\sqrt{3}}{3}$.
Tips
A common mistake is incorrectly multiplying or simplifying radicals. For example, incorrectly simplifying $\sqrt{2} \cdot \sqrt{3}$ or not rationalizing the denominator at the end. Another common mistake may be not reducing the fraction completely.
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