Certainly! Let's determine which of the given expressions is not equivalent to [tex]\( a^{\frac{1}{4}} \)[/tex].
### Expression A: [tex]\( \left(a^{\frac{1}{8}}\right)^2 \)[/tex]
We can simplify this expression by applying the power of a power rule:
[tex]\[
\left(a^{\frac{1}{8}}\right)^2 = a^{\frac{1}{8} \cdot 2} = a^{\frac{2}{8}} = a^{\frac{1}{4}}
\][/tex]
So, Expression A is equivalent to [tex]\( a^{\frac{1}{4}} \)[/tex].
### Expression B: [tex]\( a^{\frac{3}{4}} \div a^{\frac{1}{2}} \)[/tex]
We can simplify this by applying the quotient of powers rule (subtract the exponents):
[tex]\[
a^{\frac{3}{4}} \div a^{\frac{1}{2}} = a^{\frac{3}{4} - \frac{1}{2}} = a^{\frac{3}{4} - \frac{2}{4}} = a^{\frac{1}{4}}
\][/tex]
So, Expression B is equivalent to [tex]\( a^{\frac{1}{4}} \)[/tex].
### Expression C: [tex]\( \sqrt{a} \)[/tex]
We can express this in exponent form:
[tex]\[
\sqrt{a} = a^{\frac{1}{2}}
\][/tex]
This is not equivalent to [tex]\( a^{\frac{1}{4}} \)[/tex].
### Expression D: [tex]\( a^{\frac{1}{8}} \times a^{\frac{1}{8}} \)[/tex]
We can simplify this by applying the product of powers rule (add the exponents):
[tex]\[
a^{\frac{1}{8}} \times a^{\frac{1}{8}} = a^{\frac{1}{8} + \frac{1}{8}} = a^{\frac{2}{8}} = a^{\frac{1}{4}}
\][/tex]
So, Expression D is equivalent to [tex]\( a^{\frac{1}{4}} \)[/tex].
### Conclusion
The expression that is not equivalent to [tex]\( a^{\frac{1}{4}} \)[/tex] is:
[tex]\[
\boxed{\text{C}}
\][/tex]
