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Given the complex nature and possible confusion of the original content, I will structure it clearly to ensure readability and coherence.

1. The following matrix and arithmetic operations are presented:
[tex]\[
\begin{array}{rrr}
1900 & 2 \sqrt{100} & \frac{3}{256} \\
-557 & +150 & -145 \\
& \pm 200 &
\end{array}
\][/tex]

2. Perform the long division and addition operations shown:
[tex]\[
\begin{array}{rr}
4 \, \overline{\smash{\, 6 \, 7}} & 5 \, \overline{\smash{\, 9 \, 7 \, 4 \, 2 \, 8}} \\
+34 & \\
+25 & \\
\hline
\end{array}
\][/tex]

3. Evaluate the numerical expressions:
[tex]\[
6.432 \quad 112
\][/tex]

Note: Please provide specific instructions if further clarity or context is needed.



Answer :

GinnyAnswer
Certainly! Let's break down each part of the problem step-by-step, based on the given data:

1. Evaluate the components of the matrix:

- For the element [tex]\(2 \sqrt{100}\)[/tex]:
[tex]\[ 2 \sqrt{100} = 2 \times 10 = 20 \][/tex]

- For the element [tex]\(3 / 256\)[/tex]:
[tex]\[ \frac{3}{256} \approx 0.01171875 \][/tex]

2. Construct the matrix with these evaluated elements:
[tex]\[ \begin{bmatrix} 1900 & 20 & 0.01171875 \\ -557 & 150 & -145 \\ & \pm 200 & \end{bmatrix} \][/tex]
Here, [tex]\(\pm 200\)[/tex] suggests a symmetric value around zero, which we'll denote simply as [tex]\(200\)[/tex].

3. Evaluate the division operations:

- For [tex]\(\frac{4}{67}\)[/tex]:
[tex]\[ \frac{4}{67} \approx 0.05970149253731343 \][/tex]

- For [tex]\(\frac{5}{97428}\)[/tex]:
[tex]\[ \frac{5}{97428} \approx 0.0000513199490906105 \][/tex]

4. Evaluate the summation:
[tex]\[ 34 + 25 = 59 \][/tex]

5. Collecting the results:
- The matrix after evaluation:
[tex]\[ \begin{bmatrix} 1900 & 20 & 0.01171875 \\ -557 & 150 & -145 \\ & 200 & \end{bmatrix} \][/tex]

- The results of the division operations:
[tex]\[ 0.05970149253731343 \quad \text{and} \quad 0.0000513199490906105 \][/tex]

- Summation result:
[tex]\[ 59 \][/tex]

- Given results:
[tex]\[ 6.432 \quad \text{and} \quad 112 \][/tex]

Combining all these, we have our final results:

- [tex]\(2 \sqrt{100} = 20\)[/tex]
- [tex]\(3 / 256 \approx 0.01171875\)[/tex]
- [tex]\(\frac{4}{67} \approx 0.05970149253731343\)[/tex]
- [tex]\(\frac{5}{97428} \approx 0.0000513199490906105\)[/tex]
- Summation of [tex]\(34 + 25 = 59\)[/tex]
- Matrix:
[tex]\[ \begin{bmatrix} 1900 & 20 & 0.01171875 \\ -557 & 150 & -145 \\ & 200 & \end{bmatrix} \][/tex]
- Final given values:
[tex]\[ 6.432 \quad \text{and} \quad 112 \][/tex]

Thus, each operation is evaluated as shown, providing the comprehensive and detailed solution required.

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