Simplify the expression $\frac{a^{\frac{3}{20}}}{a^{\frac{1}{5}}}$ and express the answer with positive exponents.

1. Problem Description

Simplify the expression $\frac{a^{\frac{3}{20}}}{a^{\frac{1}{5}}}$ and express the answer with positive exponents.

2. Solution Steps

To simplify the expression $\frac{a^{\frac{3}{20}}}{a^{\frac{1}{5}}}$, we can use the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. In this case, $m = \frac{3}{20}$ and $n = \frac{1}{5}$.

So we have

$$\frac{a^{\frac{3}{20}}}{a^{\frac{1}{5}}} = a^{\frac{3}{20} - \frac{1}{5}}$$

To subtract the exponents, we need a common denominator. Since $5 \times 4 = 20$, we can rewrite $\frac{1}{5}$ as $\frac{4}{20}$. Then we have

$$a^{\frac{3}{20} - \frac{4}{20}} = a^{\frac{3-4}{20}} = a^{\frac{-1}{20}}$$

Since we need to express the answer with positive exponents, we can rewrite $a^{\frac{-1}{20}}$ as $\frac{1}{a^{\frac{1}{20}}}$.

3. Final Answer

$\frac{1}{a^{\frac{1}{20}}}$