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Let's work through simplifying the given expression.

The expression is:

\[
\left( \frac{\sqrt[4]{a}}{a^{-3} \cdot a^{\frac{5}{4}}} \right)^{\frac{1}{4}}
\]

### Step 1: Simplify the base inside the parentheses

We first simplify the term inside the parentheses. The numerator is \( \sqrt[4]{a} \), which can be written as \( a^{\frac{1}{4}} \).

Thus, the expression becomes:

\[
\left( \frac{a^{\frac{1}{4}}}{a^{-3} \cdot a^{\frac{5}{4}}} \right)^{\frac{1}{4}}
\]

### Step 2: Combine the powers of \( a \) in the denominator

In the denominator, we have \( a^{-3} \cdot a^{\frac{5}{4}} \). Using the property of exponents \( a^m \cdot a^n = a^{m+n} \), we add the exponents:

\[
a^{-3} \cdot a^{\frac{5}{4}} = a^{-3 + \frac{5}{4}} = a^{-\frac{12}{4} + \frac{5}{4}} = a^{-\frac{7}{4}}
\]

### Step 3: Simplify the fraction

Now the expression becomes:

\[
\left( \frac{a^{\frac{1}{4}}}{a^{-\frac{7}{4}}} \right)^{\frac{1}{4}}
\]

Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we subtract the exponents:

\[
a^{\frac{1}{4} - \left(-\frac{7}{4}\right)} = a^{\frac{1}{4} + \frac{7}{4}} = a^{\frac{8}{4}} = a^2
\]

### Step 4: Apply the outer exponent

We now apply the outer exponent \( \frac{1}{4} \):

\[
(a^2)^{\frac{1}{4}} = a^{2 \cdot \frac{1}{4}} = a^{\frac{1}{2}}
\]

### Final Answer:

Thus, the simplified expression is:

\[
a^{\frac{1}{2}}
\]

Would you like more details or have any questions?

### 5 Related Questions:
1. How do you handle fractional exponents and roots in different bases?
2. What are the key properties of exponents used in simplifying expressions?
3. Can you explain how to multiply and divide terms with exponents?
4. What happens if the base \( a \) is negative or zero in these types of problems?
5. How can you express roots like cube roots or fourth roots as fractional exponents?

### Tip:
Always simplify exponents in stages, combining terms carefully using exponent rules before applying outer operations.

Assume a > 0. Simplify the expression ((4th root of a) / (a^(-3) * a^(5/4)))^(1/4) to the form a^n where n is a rational number in its lowest terms.

Learn how to simplify a complex expression with rational exponents. This step-by-step solution involves the rules of exponents to simplify the given expression (4th root of a) / (a^(-3) * a^(5/4)) to a form like a^n, where n is a rational number.

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