The problem is to simplify the expression $\sqrt[4]{81a^{16}b^4}$. It is assumed that all variables represent positive real numbers. We need to simplify the expression using integers or fractions for any numbers in the expression, and using positive exponents only.

1. Problem Description

The problem is to simplify the expression $\sqrt[4]{81a^{16}b^4}$. It is assumed that all variables represent positive real numbers. We need to simplify the expression using integers or fractions for any numbers in the expression, and using positive exponents only.

2. Solution Steps

We are given the expression $\sqrt[4]{81a^{16}b^4}$. We can rewrite this expression using exponents:

$\sqrt[4]{81a^{16}b^4} = (81a^{16}b^4)^{1/4}$

We know that $81 = 3^4$.

Also, we have the property that $(x^m)^n = x^{m \cdot n}$. Using these facts, we can simplify the expression as follows:

$(81a^{16}b^4)^{1/4} = (3^4a^{16}b^4)^{1/4} = (3^4)^{1/4}(a^{16})^{1/4}(b^4)^{1/4}$

$= 3^{4 \cdot (1/4)}a^{16 \cdot (1/4)}b^{4 \cdot (1/4)} = 3^1a^4b^1 = 3a^4b$.

3. Final Answer

$3a^4b$