Let's break down the computation for the given expression step-by-step.
The expression we need to solve is:
[tex]\[
\sqrt{\frac{9}{89} + \frac{16}{49}} + \left(\frac{2}{5} - \frac{1}{2}\right) + \frac{7}{5}
\][/tex]
To solve this, we need to compute each part individually.
1. Calculate [tex]\(\frac{9}{89}\)[/tex]:
[tex]\[
\frac{9}{89} \approx 0.10112359550561797
\][/tex]
2. Calculate [tex]\(\frac{16}{49}\)[/tex]:
[tex]\[
\frac{16}{49} \approx 0.32653061224489793
\][/tex]
3. Add these two values:
[tex]\[
\frac{9}{89} + \frac{16}{49} \approx 0.10112359550561797 + 0.32653061224489793 = 0.4276542077505159
\][/tex]
4. Take the square root of the sum:
[tex]\[
\sqrt{0.4276542077505159} \approx 0.6539527565126673
\][/tex]
Next, compute the subtraction in the middle term:
1. Calculate [tex]\(\frac{2}{5}\)[/tex]:
[tex]\[
\frac{2}{5} = 0.4
\][/tex]
2. Calculate [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2} = 0.5
\][/tex]
3. Subtract the two values:
[tex]\[
\frac{2}{5} - \frac{1}{2} = 0.4 - 0.5 = -0.1
\][/tex]
Finally, compute the last term:
[tex]\[
\frac{7}{5} = 1.4
\][/tex]
Combine all parts together:
[tex]\[
\sqrt{\frac{9}{89} + \frac{16}{49}} + \left(\frac{2}{5} - \frac{1}{2}\right) + \frac{7}{5} \approx 0.6539527565126673 + (-0.1) + 1.4
\][/tex]
Add the values to find the final result:
[tex]\[
0.6539527565126673 - 0.1 + 1.4 = 1.9539527565126673
\][/tex]
Therefore, the result is:
[tex]\[
\boxed{1.9539527565126673}
\][/tex]
