Certainly! Let's break down the expression [tex]\(\frac{\sqrt{x+2}}{3 x}\)[/tex] step-by-step for a comprehensive understanding.
### Step 1: Understand the Expression
The given expression is:
[tex]\[
\frac{\sqrt{x+2}}{3x}
\][/tex]
where:
- [tex]\(\sqrt{x+2}\)[/tex] is the square root of [tex]\((x + 2)\)[/tex]
- [tex]\(3x\)[/tex] is three times [tex]\(x\)[/tex]
### Step 2: Simplifying the Numerator
The numerator is [tex]\(\sqrt{x+2}\)[/tex]. It indicates the square root function applied to the expression [tex]\(x+2\)[/tex]. This operation isn’t simplified any further without specific values.
### Step 3: Simplifying the Denominator
The denominator is [tex]\(3x\)[/tex]. It is already in its simplest form as three times the variable [tex]\(x\)[/tex].
### Step 4: Combine the Terms
We combine the simplified numerator and denominator into a single fraction:
[tex]\[
\frac{\sqrt{x+2}}{3x}
\][/tex]
There are no like terms to combine or cancel out between the numerator and denominator because [tex]\(\sqrt{x+2}\)[/tex] is under the square root, and [tex]\(3x\)[/tex] is a linear term.
### Step 5: Evaluate the Expression
This expression represents a fraction where:
- The numerator [tex]\(\sqrt{x+2}\)[/tex] represents the square root of the linear transformation of [tex]\(x\)[/tex].
- The denominator [tex]\(3x\)[/tex] scales the variable [tex]\(x\)[/tex] by a factor of 3.
### Conclusion
The final simplified form of the expression remains:
[tex]\[
\frac{\sqrt{x+2}}{3x}
\][/tex]
This is the most simplified form given the information available about [tex]\(x\)[/tex].
