Let's solve the given expression step by step.

Expression:

$\sqrt{\left(\frac{1}{3} + \frac{1}{6} + \frac{1}{9} - \frac{1}{12}\right)} \div \left(\frac{1}{2} + \frac{4}{5} - \frac{2}{3}\right) + \left(\frac{7}{8} - \frac{3}{4}\right) + \frac{1}{3} + \frac{1}{5}$

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Step 1: Simplify inside the square root

The terms under the square root are: $\frac{1}{3} + \frac{1}{6} + \frac{1}{9} - \frac{1}{12}$

The least common denominator (LCD) for $3, 6, 9, 12$ is $36$. Rewrite each fraction: $\frac{1}{3} = \frac{12}{36}, \quad \frac{1}{6} = \frac{6}{36}, \quad \frac{1}{9} = \frac{4}{36}, \quad \frac{1}{12} = \frac{3}{36}$

Adding and subtracting: $\frac{12}{36} + \frac{6}{36} + \frac{4}{36} - \frac{3}{36} = \frac{19}{36}$

The square root of this is: $\sqrt{\frac{19}{36}} = \frac{\sqrt{19}}{6}$

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Step 2: Simplify the denominator of the division

The terms in the denominator are: $\frac{1}{2} + \frac{4}{5} - \frac{2}{3}$

The LCD for $2, 5, 3$ is $30$. Rewrite each fraction: $\frac{1}{2} = \frac{15}{30}, \quad \frac{4}{5} = \frac{24}{30}, \quad \frac{2}{3} = \frac{20}{30}$

Adding and subtracting: $\frac{15}{30} + \frac{24}{30} - \frac{20}{30} = \frac{19}{30}$

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Step 3: Perform the division

Now divide the square root result by the denominator: $\frac{\frac{\sqrt{19}}{6}}{\frac{19}{30}} = \frac{\sqrt{19}}{6} \times \frac{30}{19} = \frac{5 \sqrt{19}}{19}$

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Step 4: Simplify $\frac{7}{8} - \frac{3}{4}$

Rewrite $\frac{3}{4}$ as $\frac{6}{8}$: $\frac{7}{8} - \frac{6}{8} = \frac{1}{8}$

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Step 5: Combine all terms

The final expression is: $\frac{5 \sqrt{19}}{19} + \frac{1}{8} + \frac{1}{3} + \frac{1}{5}$

The LCD for $8, 3, 5$ is $120$. Rewrite each fraction: $\frac{1}{8} = \frac{15}{120}, \quad \frac{1}{3} = \frac{40}{120}, \quad \frac{1}{5} = \frac{24}{120}$

Adding these: $\frac{15}{120} + \frac{40}{120} + \frac{24}{120} = \frac{79}{120}$

Add this to $\frac{5 \sqrt{19}}{19}$: $\text{Final Answer: } \frac{5 \sqrt{19}}{19} + \frac{79}{120}$

Let me know if you want further explanation or a numerical approximation. 😊

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5 Related Questions:

1. How do you simplify square roots involving fractions?
2. What is the process to find the least common denominator (LCD)?
3. How do you handle operations with multiple fractions efficiently?
4. What are some tips for simplifying complex algebraic expressions?
5. How do you convert a mixed fraction into an improper fraction?

Tip:

Always work step by step, starting with parentheses or grouped terms, to avoid errors in complex expressions.