To solve the given expression step-by-step, we will simplify and calculate each term individually, then sum them up.
Let's break down the expression term by term:
1. First term:
[tex]\[
\frac{3}{\sqrt{2 \times 2}}
\][/tex]
Simplify the denominator:
[tex]\[
\frac{3}{\sqrt{4}} = \frac{3}{2}
\][/tex]
The result is:
[tex]\[
\frac{3}{2} \approx 2.1213203435596424
\][/tex]
2. Second term:
[tex]\[
\frac{5}{\sqrt{2 \times 2 \times 3 \times 3}}
\][/tex]
Simplify the denominator:
[tex]\[
\frac{5}{\sqrt{36}} = \frac{5}{6}
\][/tex]
The result is:
[tex]\[
\frac{5}{6} \approx 0.8333333333333334
\][/tex]
3. Third term:
[tex]\[
\frac{7}{\sqrt{3 \times 3 \times 4 \times 4}}
\][/tex]
Simplify the denominator:
[tex]\[
\frac{7}{\sqrt{144}} = \frac{7}{12}
\][/tex]
The result is:
[tex]\[
\frac{7}{12} \approx 0.5833333333333334
\][/tex]
4. Fourth term:
[tex]\[
\frac{9}{\sqrt{4 \times 4 \times 5 \times 5}}
\][/tex]
Simplify the denominator:
[tex]\[
\frac{9}{\sqrt{400}} = \frac{9}{20}
\][/tex]
The result is:
[tex]\[
\frac{9}{20} = 0.45
\][/tex]
Finally, sum all the terms:
[tex]\[
2.1213203435596424 + 0.8333333333333334 + 0.5833333333333334 + 0.45
\][/tex]
The final result is:
[tex]\[
3.9879870102263095
\][/tex]
Thus, the given expression evaluates to:
[tex]\[
3.9879870102263095
\][/tex]
