To rewrite the given expression as the sum of two separate terms, we start by simplifying and splitting the original fraction. The original expression we need to simplify is:
[tex]\[ y = \frac{-3 x^7 + 4 x^{-4}}{2 x} \][/tex]
Here are the steps:
1. Split the fraction into two parts:
[tex]\[
y = \frac{-3 x^7}{2 x} + \frac{4 x^{-4}}{2 x}
\][/tex]
2. Simplify each part separately:
- For the first part:
[tex]\[
\frac{-3 x^7}{2 x} = -\frac{3}{2} \cdot \frac{x^7}{x}
\][/tex]
Simplify the powers of [tex]\(x\)[/tex]:
[tex]\[
-\frac{3}{2} \cdot x^{7-1} = -\frac{3}{2} \cdot x^6 = -\frac{3 x^6}{2}
\][/tex]
- For the second part:
[tex]\[
\frac{4 x^{-4}}{2 x} = \frac{4}{2} \cdot \frac{x^{-4}}{x}
\][/tex]
Simplify the coefficient:
[tex]\[
= 2 \cdot \frac{x^{-4}}{x}
\][/tex]
Simplify the powers of [tex]\(x\)[/tex]:
[tex]\[
= 2 \cdot x^{-4-1} = 2 \cdot x^{-5} = \frac{2}{x^5}
\][/tex]
3. Combine the simplified terms:
[tex]\[
y = -\frac{3 x^6}{2} + \frac{2}{x^5}
\][/tex]
Therefore, the simplified expression, written as the sum of two terms, is:
[tex]\[ y = -\frac{3 x^6}{2} + \frac{2}{x^5} \][/tex]
This provides a clear step-by-step rewriting of the original expression into the sum of two simpler terms.
