To simplify the given expression [tex]\( \sqrt{\frac{2\left(3 \frac{1}{3}+1 \frac{1}{6}\right)}{\frac{1}{2} \text{ of } 1 \frac{1}{2}+3 \frac{1}{4}}} \)[/tex], perform the following steps:
1. Convert Mixed Numbers to Improper Fractions:
- Convert [tex]\( 3 \frac{1}{3} \)[/tex]:
[tex]\[
3 \frac{1}{3} = 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \approx 3.3333
\][/tex]
- Convert [tex]\( 1 \frac{1}{6} \)[/tex]:
[tex]\[
1 \frac{1}{6} = 1 + \frac{1}{6} = \frac{6}{6} + \frac{1}{6} = \frac{7}{6} \approx 1.1667
\][/tex]
- Convert [tex]\( 1 \frac{1}{2} \)[/tex]:
[tex]\[
1 \frac{1}{2} = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} = 1.5
\][/tex]
- Convert [tex]\( 3 \frac{1}{4} \)[/tex]:
[tex]\[
3 \frac{1}{4} = 3 + \frac{1}{4} = \frac{12}{4} + \frac{1}{4} = \frac{13}{4} = 3.25
\][/tex]
2. Combine Numerator Terms:
[tex]\[
3 \frac{1}{3} + 1 \frac{1}{6} = \frac{10}{3} + \frac{7}{6}
\][/tex]
To add these fractions, find a common denominator:
[tex]\[
\frac{10}{3} = \frac{20}{6} \quad \text{(since 6 is the least common multiple of 3 and 6)}
\][/tex]
[tex]\[
\frac{20}{6} + \frac{7}{6} = \frac{27}{6} = 4.5 \approx 3.3333 + 1.1667 = 4.5
\][/tex]
3. Combining the Denominator Terms:
[tex]\[
\frac{1}{2} \text{ of } 1 \frac{1}{2} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} = 0.75
\][/tex]
[tex]\[
\frac{3}{4} + 3 \frac{1}{4} = \frac{3}{4} + \frac{13}{4} = \frac{16}{4} = 4 \approx 0.75 + 3.25 = 4
\][/tex]
4. Evaluate the Entire Fraction Inside the Square Root:
- Now, simplify inside the square root:
[tex]\[
\frac{2 \left( 3 \frac{1}{3} + 1 \frac{1}{6} \right)}{ \frac{1}{2} \text{ of } 1 \frac{1}{2} + 3 \frac{1}{4}} = \frac{2 \left( 4.5 \right)}{4}
\][/tex]
[tex]\[
\frac{2 \times 4.5}{4} = \frac{9}{4} = 2.25
\][/tex]
5. Calculate the Square Root:
- To find the square root of [tex]\( 2.25 \)[/tex]:
[tex]\[
\sqrt{2.25} = 1.5
\][/tex]
Hence, the simplified result of the given expression is:
[tex]\[
\boxed{1.5}
\][/tex]
