To determine which expression is equivalent to the fourth root of [tex]\(x^{10}\)[/tex], or [tex]\(\sqrt[4]{x^{10}}\)[/tex], let's analyze each given option individually.
1. Expression: [tex]\(x^2\left(\sqrt[4]{x^2}\right)\)[/tex]
Simplify [tex]\(\sqrt[4]{x^2}\)[/tex]:
[tex]\[
\sqrt[4]{x^2} = (x^2)^{1/4} = x^{2/4} = x^{1/2}
\][/tex]
Now, consider the whole expression:
[tex]\[
x^2 \cdot x^{1/2} = x^{2 + 1/2} = x^{2.5}
\][/tex]
2. Expression: [tex]\(x^{2.2}\)[/tex]
This expression is already simplified. It directly represents [tex]\(x^{2.2}\)[/tex].
3. Expression: [tex]\(x^3(\sqrt[4]{x})\)[/tex]
Simplify [tex]\(\sqrt[4]{x}\)[/tex]:
[tex]\[
\sqrt[4]{x} = x^{1/4}
\][/tex]
Now, consider the whole expression:
[tex]\[
x^3 \cdot x^{1/4} = x^{3 + 1/4} = x^{3.25}
\][/tex]
4. Expression: [tex]\(x^5\)[/tex]
This expression is already simplified. It directly represents [tex]\(x^5\)[/tex].
Let's now compare each simplified expression to [tex]\(\sqrt[4]{x^{10}}\)[/tex]:
Simplify [tex]\(\sqrt[4]{x^{10}}\)[/tex]:
[tex]\[
\sqrt[4]{x^{10}} = (x^{10})^{1/4} = x^{10/4} = x^{2.5}
\][/tex]
Comparing the results:
- [tex]\(x^2\left(\sqrt[4]{x^2}\right)\)[/tex] simplifies to [tex]\(x^{2.5}\)[/tex]
- [tex]\(x^{2.2}\)[/tex] simplifies to [tex]\(x^{2.2}\)[/tex]
- [tex]\(x^3(\sqrt[4]{x})\)[/tex] simplifies to [tex]\(x^{3.25}\)[/tex]
- [tex]\(x^5\)[/tex] simplifies to [tex]\(x^5\)[/tex]
The only expression equivalent to [tex]\(\sqrt[4]{x^{10}}\)[/tex] is [tex]\(x^2\left(\sqrt[4]{x^2}\right)\)[/tex].
Thus, the correct answer is [tex]\(x^2\left(\sqrt[4]{x^2}\right)\)[/tex]. However, given the results, none of the options provided matches [tex]\(\sqrt[4]{x^{10}}\)[/tex], indicating no expressions provided are equivalent to [tex]\(\sqrt[4]{x^{10}}\)[/tex].
