Let's break down the given expression step by step to solve it.
We want to evaluate the expression:
[tex]\[
\sqrt{\left\{\left[\left(\frac{1}{2}-\frac{1}{3}\right) \times \frac{6}{5}:\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}\right)\right] \times\left(\frac{5}{4}: \frac{15}{8}-\frac{1}{9}\right)\right\} \times \frac{5}{2}+\frac{3}{2}}
\][/tex]
Step 1: Simplify [tex]\(\left(\frac{1}{2}-\frac{1}{3}\right)\)[/tex]
[tex]\[
\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}
\][/tex]
Step 2: Evaluate [tex]\(\left(\frac{1}{2}-\frac{1}{3}\right) \times \frac{6}{5}\)[/tex]
[tex]\[
\left(\frac{1}{6}\right) \times \frac{6}{5} = \frac{1 \times 6}{6 \times 5} = \frac{6}{30} = \frac{1}{5}
\][/tex]
Step 3: Simplify [tex]\(\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}\right)\)[/tex]
Find a common denominator:
[tex]\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{6} = \frac{1}{6}
\][/tex]
[tex]\[
\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1
\][/tex]
Step 4: Evaluate [tex]\(\left(\frac{1}{5} : 1\right)\)[/tex]
[tex]\[
\frac{1}{5} : 1 = \frac{1}{5}
\][/tex]
Thus, the first term is [tex]\(\frac{1}{5}\)[/tex].
Step 5: Simplify [tex]\(\left(\frac{5}{4} : \left(\frac{15}{8} - \frac{1}{9}\right)\right)\)[/tex]
First, we find a common denominator for [tex]\(\frac{15}{8}\)[/tex] and [tex]\(\frac{1}{9}\)[/tex].
[tex]\[
\text{LCM of 8 and 9 is 72.}
\][/tex]
[tex]\[
\frac{15}{8} = \frac{15 \times 9}{8 \times 9} = \frac{135}{72}
\][/tex]
[tex]\[
\frac{1}{9} = \frac{1 \times 8}{9 \times 8} = \frac{8}{72}
\][/tex]
[tex]\[
\frac{15}{8} - \frac{1}{9} = \frac{135}{72} - \frac{8}{72} = \frac{127}{72}
\][/tex]
Step 6: Evaluate [tex]\(\left(\frac{5}{4} : \left(\frac{127}{72}\right)\right)\)[/tex]
[tex]\[
\frac{5}{4} : \frac{127}{72} = \frac{5}{4} \times \frac{72}{127} = \frac{5 \times 72}{4 \times 127} = \frac{360}{508} = \frac{180}{254} \approx 0.7086614173228347
\][/tex]
Thus, the second term is approximately [tex]\(0.7086614173228347\)[/tex].
Step 7: Multiply the two terms
[tex]\[
\left(\frac{1}{5}\right) \times 0.7086614173228347 \approx 0.14173228346456698
\][/tex]
Step 8: Multiply the result by [tex]\(\frac{5}{2}\)[/tex] and add [tex]\(\frac{3}{2}\)[/tex]
[tex]\[
0.14173228346456698 \times \frac{5}{2} = 0.14173228346456698 \times 2.5 = 0.3543307086614175
\][/tex]
[tex]\[
0.3543307086614175 + \frac{3}{2} = 0.3543307086614175 + 1.5 = 1.8543307086614175
\][/tex]
Step 9: Take the square root of the result
[tex]\[
\sqrt{1.8543307086614175} \approx 1.3617381204407173
\][/tex]
So, the final result of the expression is approximately [tex]\( \boxed{1.3617381204407173} \)[/tex].
