What is the sum?

[tex]\[
\begin{array}{c}
\frac{2}{x^2} + \frac{4}{x^2} + \frac{8}{x^2} + \frac{6}{x^2} + \frac{6}{x^4} + \frac{6}{2x^2} \\
\hline
\end{array}
\][/tex]



Answer :

To solve the sum of the given fractions, let's break down each component and simplify it appropriately. The fractions given in the problem are:

[tex]\[ \frac{2}{x^2}, \quad \frac{4}{x^2}, \quad \frac{8}{x^2}, \quad \frac{6}{x^2}, \quad \frac{6}{2x^2}, \quad \frac{6}{x^4} \][/tex]

First, notice that:
- The fractions [tex]\(\frac{2}{x^2}, \frac{4}{x^2}, \frac{8}{x^2}, \frac{6}{x^2}\)[/tex] all have a common denominator [tex]\(x^2\)[/tex].
- The fraction [tex]\(\frac{6}{2x^2}\)[/tex] can be simplified.

For the fraction [tex]\(\frac{6}{2x^2}\)[/tex]:
[tex]\[ \frac{6}{2x^2} = \frac{6 \div 2}{x^2} = \frac{3}{x^2} \][/tex]

Next, we can write down all the simplified fractions:

[tex]\[ \frac{2}{x^2}, \quad \frac{4}{x^2}, \quad \frac{8}{x^2}, \quad \frac{6}{x^2}, \quad \frac{3}{x^2}, \quad \frac{6}{x^4} \][/tex]

Notice that [tex]\(\frac{6}{x^4}\)[/tex] is in terms of [tex]\(x^4\)[/tex] instead of [tex]\(x^2\)[/tex], which may differ in influence depending on the context, though it is equivalent if [tex]\(x=1\)[/tex].

Let's add the fractions that have a common denominator ([tex]\(x^2\)[/tex]):

[tex]\[ \frac{2}{x^2} + \frac{4}{x^2} + \frac{8}{x^2} + \frac{6}{x^2} + \frac{3}{x^2} = \frac{2 + 4 + 8 + 6 + 3}{x^2} = \frac{23}{x^2} \][/tex]

The fraction [tex]\(\frac{6}{x^4}\)[/tex] does not initially fit this sum unless we reconsider each specific value but since all others integrate, it can stay as [tex]\(\frac{6}{x^4}\)[/tex].

So, the major solution components are:

[tex]\[ \left(\frac{2}{x^2}, \frac{4}{x^2}, \frac{8}{x^2}, \frac{6}{x^2}, \frac{3}{x^2}\right) = (2.0, 4.0, 8.0, 6.0, 3.0) \][/tex]

Adding these together yields:

[tex]\[ 2.0 + 4.0 + 8.0 + 6.0 + 3.0 = 23.0 \][/tex]

So, the final result for the sum of these fractions is:

[tex]\[ 23.0 \][/tex] This sum numerically confirms the calculation and integrates it all for simplified results.

If your initial problem insists on context where [tex]\(x = 1\)[/tex]:

Given
[tex]\[ \sum_{i=1}^{n} \frac{a_{i}}{b_{i}} = \frac{\sum_{i=1}^{n} a_{i}}{\sum_{i=1}^{n} b_{i}} \][/tex]

Other handled condition retains precise weight as previously discussed ensuring adjusted numeric resultant.