Let's evaluate each given expression step by step to write them with single rational exponents.
Expression A: [tex]\(\sqrt[4]{x^3}\)[/tex]
- The fourth root of [tex]\(x^3\)[/tex] can be expressed as [tex]\(x\)[/tex] raised to the power of [tex]\(3/4\)[/tex]:
[tex]\[
\sqrt[4]{x^3} = x^{3/4}
\][/tex]
Expression B: [tex]\(\frac{1}{x^{-1}}\)[/tex]
- The denominator has [tex]\(x^{-1}\)[/tex], which can be rewritten using the property of exponents that states [tex]\(\frac{1}{x^a} = x^{-a}\)[/tex]:
[tex]\[
\frac{1}{x^{-1}} = x^{-(-1)} = x^1
\][/tex]
Thus, Expression B simplifies to:
[tex]\[
x^1
\][/tex]
Expression C: [tex]\(\sqrt[10]{x^5 \cdot x^4 \cdot x^2}\)[/tex]
- First, combine the exponents in the product [tex]\(x^5 \cdot x^4 \cdot x^2\)[/tex]:
[tex]\[
x^5 \cdot x^4 \cdot x^2 = x^{5+4+2} = x^{11}
\][/tex]
- Now, take the tenth root of [tex]\(x^{11}\)[/tex], which is equivalent to raising [tex]\(x^{11}\)[/tex] to the power of [tex]\(1/10\)[/tex]:
[tex]\[
\sqrt[10]{x^{11}} = x^{11/10}
\][/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 together:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = x^1
\][/tex]
Now, let's compare the simplified forms of each expression:
- Expression A: [tex]\( x^{3/4} \)[/tex]
- Expression B: [tex]\( x^1 \)[/tex]
- Expression C: [tex]\( x^{11/10} \)[/tex]
- Expression D: [tex]\( x^1 \)[/tex]
Both Expression B and Expression D are equivalent as they simplify to [tex]\( x^1 \)[/tex]. Thus, the equivalent expressions are B and D.
To summarize:
- Expression A becomes [tex]\( x^{3/4} \)[/tex]
- Expression B becomes [tex]\( x^1 \)[/tex]
- Expression C becomes [tex]\( x^{11/10} \)[/tex]
- Expression D becomes [tex]\( x^1 \)[/tex]
Expressions [tex]\( B \)[/tex] and [tex]\( D \)[/tex] are equivalent.
