Simplify: (3 + √2) / (3√2 * (1/√3))
Understand the Problem
The question asks to simplify the expression (3 + √2) / (3√2 * (1/√3)). This involves rationalizing the denominator and simplifying the radicals.
Answer
$\frac{3\sqrt{6} + 2\sqrt{3}}{6}$
Answer for screen readers
$\frac{3\sqrt{6} + 2\sqrt{3}}{6}$
Steps to Solve
- Simplify the denominator Rewrite the denominator by multiplying the terms:
$3\sqrt{2} \cdot \frac{1}{\sqrt{3}} = \frac{3\sqrt{2}}{\sqrt{3}}$
- Rationalize the denominator Multiply both the numerator and the denominator by $\sqrt{3}$ to remove the radical from the denominator:
$\frac{3 + \sqrt{2}}{\frac{3\sqrt{2}}{\sqrt{3}}} = \frac{(3 + \sqrt{2})\sqrt{3}}{3\sqrt{2}}$
- Expand the numerator Distribute $\sqrt{3}$ in the numerator:
$\frac{3\sqrt{3} + \sqrt{6}}{3\sqrt{2}}$
- Rationalize the denominator again Multiply both the numerator and the denominator by $\sqrt{2}$ to remove the radical from the denominator:
$\frac{(3\sqrt{3} + \sqrt{6})\sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}} = \frac{3\sqrt{6} + \sqrt{12}}{3 \cdot 2}$
- Simplify the expression Simplify $\sqrt{12}$ and the denominator:
$\frac{3\sqrt{6} + 2\sqrt{3}}{6}$
$\frac{3\sqrt{6} + 2\sqrt{3}}{6}$
More Information
The expression has been simplified by rationalizing the denominator and combining like terms. We successively rationalized the denominator by multiplying by $\sqrt{3}$ and $\sqrt{2}$ to eliminate the radicals from the denominator.
Tips
A common mistake when simplifying expressions with radicals is not rationalizing the denominator completely. Another mistake is incorrectly multiplying or simplifying radicals. Ensure each radical is simplified and that the denominator does not contain any radicals.
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