Let's solve the given problem step-by-step.
We start with the expressions:
[tex]\[
\sqrt[4]{2 x^2} \quad \text{and} \quad \sqrt[4]{2 x^3}
\][/tex]
First, we convert these expressions into a more manageable form using fractional exponents:
1. The first expression [tex]\(\sqrt[4]{2 x^2}\)[/tex] can be written as:
[tex]\[
(2 x^2)^{\frac{1}{4}}
\][/tex]
2. The second expression [tex]\(\sqrt[4]{2 x^3}\)[/tex] can be written as:
[tex]\[
(2 x^3)^{\frac{1}{4}}
\][/tex]
Next, let's simplify each of these expressions individually:
For the expression [tex]\((2 x^2)^{\frac{1}{4}}\)[/tex]:
[tex]\[
(2 x^2)^{\frac{1}{4}} = (2)^{\frac{1}{4}} \cdot (x^2)^{\frac{1}{4}} = 2^{\frac{1}{4}} \cdot x^{\frac{2}{4}} = 2^{\frac{1}{4}} \cdot x^{\frac{1}{2}}
\][/tex]
For the expression [tex]\((2 x^3)^{\frac{1}{4}}\)[/tex]:
[tex]\[
(2 x^3)^{\frac{1}{4}} = (2)^{\frac{1}{4}} \cdot (x^3)^{\frac{1}{4}} = 2^{\frac{1}{4}} \cdot x^{\frac{3}{4}}
\][/tex]
Now, we multiply these two simplified expressions together:
[tex]\[
(2 x^2)^{\frac{1}{4}} \cdot (2 x^3)^{\frac{1}{4}} = \left(2^{\frac{1}{4}} \cdot x^{\frac{1}{2}}\right) \cdot \left(2^{\frac{1}{4}} \cdot x^{\frac{3}{4}}\right)
\][/tex]
Combining the constants and exponents, we have:
[tex]\[
2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot x^{\frac{1}{2}} \cdot x^{\frac{3}{4}} = 2^{\frac{1}{4} + \frac{1}{4}} \cdot x^{\frac{1}{2} + \frac{3}{4}}
\][/tex]
Simplifying the exponents, we get:
[tex]\[
2^{\frac{2}{4}} \cdot x^{\frac{2}{4} + \frac{3}{4}} = 2^{\frac{1}{2}} \cdot x^{\frac{5}{4}}
\][/tex]
Finally, simplify further:
[tex]\[
2^{\frac{1}{2}} = \sqrt{2}
\][/tex]
Thus, the equivalent expression is:
[tex]\[
\sqrt{2} \cdot x^{\frac{5}{4}}
\][/tex]
Numerically, [tex]\(\sqrt{2} \approx 1.41421356237309\)[/tex]. Therefore, the final expression is:
[tex]\[
1.41421356237309 \cdot x^{\frac{5}{4}}
\][/tex]
So, [tex]\(\sqrt[4]{2 x^2} \cdot \sqrt[4]{2 x^3}\)[/tex] is equivalent to:
[tex]\[
1.41421356237309 \cdot (x^2)^{\frac{1}{4}} \cdot (x^3)^{\frac{1}{4}} = 1.41421356237309 \cdot x^{\frac{5}{4}}
\][/tex]
