Let's simplify each of the given expressions step by step.
### Expression 1: [tex]\( \sqrt[4]{\frac{3}{2x}} \)[/tex]
To simplify this expression, we recognize that the fourth root of a fraction is the fraction of the fourth roots of the numerator and denominator:
[tex]\[
\sqrt[4]{\frac{3}{2x}} = \frac{\sqrt[4]{3}}{\sqrt[4]{2x}}
\][/tex]
We can simplify further:
[tex]\[
\frac{\sqrt[4]{3}}{\sqrt[4]{2}\cdot\sqrt[4]{x}} = \frac{\sqrt[4]{3}}{\sqrt[4]{2} x^{1/4}}
\][/tex]
Thus, the simplified form is:
[tex]\[
\frac{2^{3/4} \cdot 3^{1/4}}{2 \cdot x^{1/4}} = \frac{2^{3/4} \cdot 3^{1/4}}{2 \cdot x^{1/4}}
\][/tex]
### Expression 2: [tex]\( \frac{\sqrt[4]{6x}}{2x} \)[/tex]
Here, we simplify the numerator first and then divide by the denominator:
[tex]\[
\frac{\sqrt[4]{6x}}{2x} = \frac{\sqrt[4]{6}\sqrt[4]{x}}{2x}
\][/tex]
Simplified further, we get:
[tex]\[
\frac{6^{1/4}x^{1/4}}{2x} = \frac{6^{1/4}}{2 x^{3/4}}
\][/tex]
### Expression 3: [tex]\( \frac{\sqrt[4]{24x^3}}{2x} \)[/tex]
To simplify, first simplify the numerator, then divide by the denominator:
[tex]\[
\frac{\sqrt[4]{24x^3}}{2x} = \frac{\sqrt[4]{24} \cdot \sqrt[4]{x^3}}{2x}
\][/tex]
Simplify further:
[tex]\[
\frac{24^{1/4} \cdot (x^3)^{1/4}}{2x} = \frac{24^{1/4} \cdot x^{3/4}}{2x} = \frac{24^{1/4} \cdot x^{3/4}}{2 \cdot x^{1}} = \frac{24^{1/4}}{2 \cdot x^{1 - 3/4}} = \frac{24^{1/4}}{2 \cdot x^{1/4}}
\][/tex]
### Expression 4: [tex]\( \frac{\sqrt[4]{24x^3}}{16x^4} \)[/tex]
Simplify the numerator first, then divide by the denominator:
[tex]\[
\frac{\sqrt[4]{24x^3}}{16x^4} = \frac{\sqrt[4]{24} \cdot \sqrt[4]{x^3}}{16 x^4}
\][/tex]
Simplify further:
[tex]\[
\frac{24^{1/4} \cdot x^{3/4}}{16 \cdot x^4} = \frac{24^{1/4} \cdot x^{3/4}}{16 \cdot x^{4}} = \frac{24^{1/4}}{16 \cdot x^{4 - 3/4}} = \frac{24^{1/4}}{16 \cdot x^{16/4 - 3/4}} = \frac{24^{1/4}}{16 \cdot x^{13/4}}
\][/tex]
### Expression 5: [tex]\( \sqrt[4]{12x^2} \)[/tex]
Simplify directly by applying the property of the fourth root:
[tex]\[
\sqrt[4]{12x^2} = \sqrt[4]{12} \cdot \sqrt[4]{x^2}
\][/tex]
Simplify further:
[tex]\[
12^{1/4} \cdot (x^2)^{1/4} = 12^{1/4} \cdot x^{2/4} = 12^{1/4} \cdot x^{1/2}
\][/tex]
Putting everything together, the simplified forms of the expressions are:
1. [tex]\( \frac{2^{3/4} \cdot 3^{1/4}}{2 \cdot x^{1/4}} \)[/tex]
2. [tex]\( \frac{6^{1/4}}{2 x^{3/4}} \)[/tex]
3. [tex]\( \frac{24^{1/4}}{2 x^{1/4}} \)[/tex]
4. [tex]\( \frac{24^{1/4}}{16 x^{13/4}} \)[/tex]
5. [tex]\( 12^{1/4} \cdot x^{1/2} \)[/tex]
These expressions represent the simplified forms of the given mathematical expressions.
