We are asked to simplify the expression $\frac{a^{\frac{1}{2}} \times a^{\frac{1}{4}}}{a}$.

1. Problem Description

We are asked to simplify the expression $\frac{a^{\frac{1}{2}} \times a^{\frac{1}{4}}}{a}$.

2. Solution Steps

First, we simplify the numerator. When multiplying terms with the same base, we add the exponents. So, $a^{\frac{1}{2}} \times a^{\frac{1}{4}} = a^{\frac{1}{2} + \frac{1}{4}}$. To add the fractions, we need a common denominator, which is

4. Thus, $\frac{1}{2} = \frac{2}{4}$, and $\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$. Therefore, $a^{\frac{1}{2}} \times a^{\frac{1}{4}} = a^{\frac{3}{4}}$.

Now we have the expression $\frac{a^{\frac{3}{4}}}{a}$. Remember that $a$ can be written as $a^1$. When dividing terms with the same base, we subtract the exponents. Thus, $\frac{a^{\frac{3}{4}}}{a^1} = a^{\frac{3}{4} - 1}$.

We need to subtract 1 from $\frac{3}{4}$. We can write 1 as $\frac{4}{4}$. So, $\frac{3}{4} - 1 = \frac{3}{4} - \frac{4}{4} = \frac{3-4}{4} = \frac{-1}{4} = -\frac{1}{4}$.

Therefore, $\frac{a^{\frac{3}{4}}}{a} = a^{-\frac{1}{4}}$.

We can also write this as $\frac{1}{a^{\frac{1}{4}}}$.

3. Final Answer

$\frac{1}{a^{\frac{1}{4}}}$