Let's break down and simplify the given mathematical expression step-by-step:
Given expression:
[tex]$(\sqrt{x})^{-\frac{2}{3}} \sqrt{y^4}-\sqrt{x y y^{1 / 2}}$[/tex]
### Step 1: Simplify each part of the expression individually.
#### Part 1: Simplify [tex]\((\sqrt{x})^{-\frac{2}{3}}\)[/tex]
The expression [tex]\(\sqrt{x}\)[/tex] can be written as [tex]\(x^{1/2}\)[/tex].
Therefore, [tex]\((\sqrt{x})^{-\frac{2}{3}}\)[/tex] becomes:
[tex]$(x^{1/2})^{-\frac{2}{3}}$[/tex]
Applying the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]$x^{\left( \frac{1}{2} \cdot -\frac{2}{3} \right)} = x^{-\frac{2}{6}} = x^{-\frac{1}{3}}$[/tex]
So, [tex]\((\sqrt{x})^{-\frac{2}{3}} = x^{-\frac{1}{3}}\)[/tex].
#### Part 2: Simplify [tex]\(\sqrt{y^4}\)[/tex]
The expression [tex]\(\sqrt{y^4}\)[/tex] can be written as:
[tex]$y^{4/2} = y^2$[/tex]
Thus, [tex]\(\sqrt{y^4} = y^2\)[/tex].
#### Part 3: Simplify [tex]\(\sqrt{x y y^{1/2}}\)[/tex]
First, rewrite the expression under the square root:
[tex]$\sqrt{x y y^{1/2}} = \sqrt{x \cdot y \cdot y^{1/2}}$[/tex]
Combine the [tex]\(y\)[/tex] terms:
[tex]$= \sqrt{x \cdot y^{1 + 1/2}} = \sqrt{x \cdot y^{3/2}}$[/tex]
So, [tex]\(\sqrt{x y y^{1/2}} = \sqrt{x y^{3/2}}\)[/tex].
### Step 2: Combine and simplify the individual parts.
Now we combine the simplified parts from Step 1 into the original expression:
[tex]$(x^{-\frac{1}{3}} y^2) - \sqrt{x y^{3/2}}$[/tex]
#### Simplify the combined expression:
1. The first part:
[tex]$x^{-\frac{1}{3}} y^2$[/tex]
2. The second part:
[tex]$\sqrt{x y^{3/2}}$[/tex]
We can't directly combine these terms further, but we can express them together:
[tex]$x^{-\frac{1}{3}} y^2 - \sqrt{x y^{3/2}}$[/tex]
So, the simplified form of the given expression is:
[tex]$x^{-\frac{1}{3}} y^2 - \sqrt{x y^{3/2}}$[/tex]
