Sure, let's simplify the given expression step-by-step:
The expression we start with is:
[tex]\[ \sqrt{x} + 2 \sqrt{x^3} + 3 x \sqrt{x} - \frac{1}{x} \sqrt{8 x^3} \][/tex]
First, recall that [tex]\(\sqrt{x^3} = \sqrt{x \cdot x^2} = x \sqrt{x}\)[/tex]. Using this, we can rewrite parts of the expression:
- [tex]\(2 \sqrt{x^3}\)[/tex] becomes [tex]\(2 \cdot x \sqrt{x}\)[/tex].
- [tex]\(3 x \sqrt{x}\)[/tex] remains as it is.
- [tex]\(\frac{1}{x} \sqrt{8 x^3}\)[/tex] will need more simplification. Note that [tex]\(\sqrt{8 x^3} = \sqrt{8} \sqrt{x^3} = 2 \sqrt{2} x \sqrt{x}\)[/tex].
Now, rewriting the expression using these simplifications:
[tex]\[ \sqrt{x} + 2(x \sqrt{x}) + 3 x \sqrt{x} - \frac{1}{x} (2 \sqrt{2} x \sqrt{x}) \][/tex]
Next, simplify the term [tex]\(\frac{1}{x} (2 \sqrt{2} x \sqrt{x})\)[/tex]:
[tex]\[ \frac{1}{x} \cdot 2 \sqrt{2} x \sqrt{x} = 2 \sqrt{2} \sqrt{x} \][/tex]
So, we now have:
[tex]\[ \sqrt{x} + 2 x \sqrt{x} + 3 x \sqrt{x} - 2 \sqrt{2} \sqrt{x} \][/tex]
Now, combine the like terms. Notice that we have terms involving [tex]\(\sqrt{x}\)[/tex] and terms involving [tex]\(x \sqrt{x}\)[/tex]:
1. [tex]\(\sqrt{x}\)[/tex] terms:
- [tex]\(\sqrt{x}\)[/tex]
- [tex]\(-2 \sqrt{2} \sqrt{x}\)[/tex]
Combine them:
[tex]\[ \sqrt{x} - 2 \sqrt{2} \sqrt{x} \][/tex]
2. [tex]\(x \sqrt{x}\)[/tex] terms:
- [tex]\(2 x \sqrt{x}\)[/tex]
- [tex]\(3 x \sqrt{x}\)[/tex]
Combine them:
[tex]\[ (2 + 3) x \sqrt{x} = 5 x \sqrt{x} \][/tex]
So, putting everything together, we get the simplified form of the expression:
[tex]\[ 3 x^{3/2} + \sqrt{x} + 2 \sqrt{x^3} - 2 \sqrt{2} \sqrt{x^3} / x \][/tex]
We have successfully simplified the expression as:
[tex]\[ 3 x^{3/2} + \sqrt{x} + 2 \sqrt{x^3} - 2 \sqrt{2} \sqrt{x^3} / x \][/tex]
