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

AlgebraExponentsSimplificationFractionsAlgebraic Expressions
2025/3/8

1. Problem Description

We are asked to simplify the expression a12×a14a\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, a12×a14=a12+14a^{\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 a34a\frac{a^{\frac{3}{4}}}{a}. Remember that aa can be written as a1a^1. When dividing terms with the same base, we subtract the exponents. Thus, a34a1=a341\frac{a^{\frac{3}{4}}}{a^1} = a^{\frac{3}{4} - 1}.
We need to subtract 1 from 34\frac{3}{4}. We can write 1 as 44\frac{4}{4}. So, 341=3444=344=14=14\frac{3}{4} - 1 = \frac{3}{4} - \frac{4}{4} = \frac{3-4}{4} = \frac{-1}{4} = -\frac{1}{4}.
Therefore, a34a=a14\frac{a^{\frac{3}{4}}}{a} = a^{-\frac{1}{4}}.
We can also write this as 1a14\frac{1}{a^{\frac{1}{4}}}.

3. Final Answer

1a14\frac{1}{a^{\frac{1}{4}}}

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