To solve the expression [tex]\(\sqrt{2:\left\{\frac{5}{6}-\left[\frac{2}{3}+\left(\frac{1}{2}\right)^4:\left(\frac{1}{2}\right)^3-\frac{5}{6}\right]+\frac{3}{4}\right\}:\left\{\frac{2}{3}-\left[\frac{4}{5}+3-\left(\frac{4}{3}+2\right)\right]\right\}+1}\)[/tex], let's break it down step by step.
1. Simplify the innermost parentheses:
[tex]\[
\left(\frac{1}{2}\right)^4 : \left(\frac{1}{2}\right)^3 - \frac{5}{6}
\][/tex]
Simplify [tex]\(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\)[/tex] and [tex]\(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\)[/tex].
[tex]\[
\frac{1}{16} : \frac{1}{8} = \frac{1}{16} \times 8 = \frac{1}{2}
\][/tex]
So,
[tex]\[
\frac{1}{2} - \frac{5}{6}
\][/tex]
To perform the subtraction, find a common denominator (6 in this case):
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
Thus,
[tex]\[
\frac{3}{6} - \frac{5}{6} = -\frac{2}{6} = -\frac{1}{3}
\][/tex]
Adding [tex]\(\frac{2}{3}\)[/tex] to this:
[tex]\[
\frac{2}{3} + \left(-\frac{1}{3}\right) = \frac{2}{3} - \frac{1}{3} = \frac{1}{3}
\][/tex]
Therefore, the result of the innermost parentheses is:
[tex]\[
\frac{1}{3}
\][/tex]
2. Simplify the next bracket:
[tex]\[
\frac{5}{6} - \frac{1}{3} + \frac{3}{4}
\][/tex]
To perform these operations, convert everything to a common denominator (12):
[tex]\[
\frac{5}{6} = \frac{10}{12}, \quad \frac{1}{3} = \frac{4}{12}, \quad \frac{3}{4} = \frac{9}{12}
\][/tex]
So,
[tex]\[
\frac{10}{12} - \frac{4}{12} + \frac{9}{12}
\][/tex]
Simplify in steps:
[tex]\[
\frac{10}{12} - \frac{4}{12} = \frac{6}{12} = \frac{1}{2}
\][/tex]
Then,
[tex]\[
\frac{1}{2} + \frac{9}{12} = \frac{6}{12} + \frac{9}{12} = \frac{15}{12} = \frac{5}{4} = 1.25
\][/tex]
3. Simplify the denominator:
[tex]\[
\frac{2}{3} - \left[ \frac{4}{5} + 3 - \left( \frac{4}{3} + 2 \right) \right]
\][/tex]
Simplify the inner expression:
[tex]\[
\frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3}
\][/tex]
Thus,
[tex]\[
\frac{4}{5} + 3 - \frac{10}{3}
\][/tex]
Convert everything to a common denominator (15):
[tex]\[
\frac{4}{5} = \frac{12}{15}, \quad 3 = \frac{45}{15}, \quad \frac{10}{3} = \frac{50}{15}
\][/tex]
So,
[tex]\[
\frac{12}{15} + \frac{45}{15} - \frac{50}{15} = \frac{57}{15} - \frac{50}{15} = \frac{7}{15}
\][/tex]
Therefore, the expression becomes:
[tex]\[
\frac{2}{3} - \frac{7}{15}
\][/tex]
Convert to a common denominator (15):
[tex]\[
\frac{2}{3} = \frac{10}{15}
\][/tex]
Then,
[tex]\[
\frac{10}{15} - \frac{7}{15} = \frac{3}{15} = \frac{1}{5} = 0.2
\][/tex]
4. Calculate the expression inside the square root:
[tex]\[
\frac{1.25}{0.2} = 6.25
\][/tex]
Adding 1:
[tex]\[
6.25 + 1 = 7.25
\][/tex]
5. Calculate the outer fraction:
[tex]\[
\frac{2}{7.25} \approx 0.27586
\][/tex]
6. Finally, take the square root:
[tex]\[
\sqrt{0.27586} \approx 0.52523
\][/tex]
Therefore, the final result of the expression is approximately [tex]\(0.52523\)[/tex].
