To determine which expression is equivalent to [tex]\(\frac{1}{2 x^{\frac{5}{2}}}\)[/tex], let's carefully analyze and transform it step by step.
1. Start with the given expression:
[tex]\[
\frac{1}{2 x^{\frac{5}{2}}}
\][/tex]
2. Recognize that any expression of the form [tex]\(x^a\)[/tex] can be represented with the exponent inside the denominator. Specifically, [tex]\(x^{\frac{5}{2}}\)[/tex] represents [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{5}{2}\)[/tex].
3. Substitute this into the expression:
[tex]\[
\frac{1}{2 x^{\frac{5}{2}}}
\][/tex]
4. This can be interpreted as having the term [tex]\(2 x^{\frac{5}{2}}\)[/tex] in the denominator.
5. Convert the exponent [tex]\(\frac{5}{2}\)[/tex] to a decimal representation which is [tex]\(2.5\)[/tex]:
[tex]\[
\frac{1}{2 x^{2.5}}
\][/tex]
Thus, an equivalent form of the given expression is:
[tex]\[
\frac{1}{2 x^{2.5}}
\][/tex]
Therefore, [tex]\(\frac{1}{2 x^{2.5}}\)[/tex] is indeed equivalent to [tex]\(\frac{1}{2 x^{\frac{5}{2}}}\)[/tex].
