Let's simplify each given expression step-by-step:
### Expression A: [tex]\(\sqrt[4]{x^3}\)[/tex]
The fourth root of [tex]\(x^3\)[/tex] can be written with a rational exponent:
[tex]\[
\sqrt[4]{x^3} = x^{\frac{3}{4}}
\][/tex]
### Expression B: [tex]\(\frac{1}{x^{-1}}\)[/tex]
First, recall that the negative exponent rule states [tex]\(x^{-a} = \frac{1}{x^a}\)[/tex]. Using this, we get:
[tex]\[
\frac{1}{x^{-1}} = x^{1}
\][/tex]
### Expression C: [tex]\(\sqrt[10]{x^5 \cdot x^4 \cdot x^2}\)[/tex]
Combine the exponents inside the root:
[tex]\[
x^5 \cdot x^4 \cdot x^2 = x^{5+4+2} = x^{11}
\][/tex]
Then, apply the tenth root:
[tex]\[
\sqrt[10]{x^{11}} = x^{\frac{11}{10}}
\][/tex]
### Expression D: [tex]\(x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}}\)[/tex]
Combine the exponents by adding them:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = x^{1} = x
\][/tex]
### Summary and Equivalence Check
- Expression A: [tex]\(x^{\frac{3}{4}}\)[/tex]
- Expression B: [tex]\(x\)[/tex]
- Expression C: [tex]\(x^{\frac{11}{10}}\)[/tex]
- Expression D: [tex]\(x\)[/tex]
By comparing the simplified forms with each other:
1. [tex]\(x^{\frac{3}{4}}\)[/tex]
2. [tex]\(x\)[/tex]
3. [tex]\(x^{\frac{11}{10}}\)[/tex]
4. [tex]\(x\)[/tex]
Thus, expressions B and D are equivalent because both simplify to [tex]\(x\)[/tex]. Expressions A and C are not equivalent to any other expressions in this set.
