$\frac{2 \sqrt{2}}{\sqrt{6}}+\frac{1}{\sqrt{3}} ?
Answer (2)
The given expression simplifies to 3 . This involves simplifying each term and then combining them together. The steps include rationalizing the denominators and adding the resulting fractions.
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Simplify the first term: 6 2 2 = 3 2
Rationalize the denominators: 3 2 = 3 2 3 and 3 1 = 3 3
Add the simplified terms: 3 2 3 + 3 3 = 3 3 3
Simplify the final expression: 3 3 3 = 3 . The final answer is 3
Explanation
Understanding the Problem We are asked to simplify the expression 6 2 2 + 3 1 . This involves simplifying radicals and adding the resulting terms.
Simplifying the First Term First, let's simplify the term 6 2 2 . We can rewrite 6 as 2 ⋅ 3 . Thus, we have 6 2 2 = 2 ⋅ 3 2 2 = 3 2
Rationalizing the Denominators Now, let's rationalize the denominator of both terms. For the first term, we have 3 2 = 3 2 ⋅ 3 3 = 3 2 3 For the second term, we have 3 1 = 3 1 ⋅ 3 3 = 3 3
Adding the Terms Now, we add the two simplified terms: 3 2 3 + 3 3 = 3 2 3 + 3 = 3 3 3 = 3
Final Answer Therefore, the simplified expression is 3 .
Examples
Radical expressions like this can appear when calculating distances in geometry, especially when dealing with triangles and the Pythagorean theorem. For example, if you have a right triangle with certain side lengths involving square roots, simplifying such expressions can help you find the exact length of another side or the area of the triangle. Simplifying radical expressions is also useful in physics, such as when calculating the period of a pendulum or dealing with wave functions in quantum mechanics.
