Sure, let's go through the expression step by step and simplify it.
We start with:
[tex]\[ 558 - 3 \left(\frac{5}{\sqrt{2}}\right) + 5\sqrt{80} - 2\left(\frac{3}{2}\right) \][/tex]
Step 1: Simplify [tex]\(\frac{5}{\sqrt{2}}\)[/tex]:
[tex]\[ \frac{5}{\sqrt{2}} \approx 3.5355339059327373 \][/tex]
Step 2: Multiply by 3:
[tex]\[ 3 \left(\frac{5}{\sqrt{2}}\right) = 3 \times 3.5355339059327373 \approx 10.606601717798211 \][/tex]
Step 3: Simplify [tex]\(\sqrt{80}\)[/tex]:
[tex]\[ \sqrt{80} \approx 8.94427190999916 \][/tex]
Step 4: Multiply by 5:
[tex]\[ 5\sqrt{80} = 5 \times 8.94427190999916 \approx 44.721359549995796 \][/tex]
Step 5: Simplify [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[ \frac{3}{2} = 1.5 \][/tex]
Step 6: Multiply by 2:
[tex]\[ 2 \left(\frac{3}{2}\right) = 2 \times 1.5 = 3.0 \][/tex]
Step 7: Substitute all simplified terms back into the expression:
[tex]\[ 558 - 10.606601717798211 + 44.721359549995796 - 3.0 \][/tex]
Step 8: Combine all terms:
[tex]\[ 558 - 10.606601717798211 + 44.721359549995796 - 3.0 \approx 589.1147578321976 \][/tex]
So the simplified value of the original expression is approximately:
[tex]\[ 589.1147578321976 \][/tex]
