Certainly! Let's simplify the given expression [tex]\(\frac{x + 5 x^3}{\sqrt{x}}\)[/tex].
First, we can break the fraction into two separate terms:
[tex]\[ \frac{x + 5 x^3}{\sqrt{x}} = \frac{x}{\sqrt{x}} + \frac{5 x^3}{\sqrt{x}}. \][/tex]
Now simplify each term individually.
1. Simplify the first term [tex]\(\frac{x}{\sqrt{x}}\)[/tex]:
[tex]\[ \frac{x}{\sqrt{x}} = \frac{x}{x^{1/2}} = x^{1 - 1/2} = x^{1/2}. \][/tex]
2. Simplify the second term [tex]\(\frac{5 x^3}{\sqrt{x}}\)[/tex]:
[tex]\[ \frac{5 x^3}{\sqrt{x}} = \frac{5 x^3}{x^{1/2}} = 5 x^{3 - 1/2} = 5 x^{5/2}. \][/tex]
So, combining these results, we get:
[tex]\[ \frac{x + 5 x^3}{\sqrt{x}} = x^{1/2} + 5 x^{5/2}. \][/tex]
In this form, [tex]\(m = \frac{1}{2}\)[/tex] and [tex]\(n = \frac{5}{2}\)[/tex]. Therefore, the simplified expression is:
[tex]\[ x^{1/2} + 5 x^{5/2}. \][/tex]
