The expression provided is:

$\left(b^2 \cdot a^{-\frac{1}{4}}\right)^{\frac{3}{4}}$

We need to simplify this expression while avoiding negative exponents.

Step-by-step simplification:

1. Apply the outer exponent: Use the property of exponents $(x^m)^n = x^{m \cdot n}$ to distribute the $\frac{3}{4}$ to both $b^2$ and $a^{-\frac{1}{4}}$.

   $\left(b^2 \cdot a^{-\frac{1}{4}}\right)^{\frac{3}{4}} = b^{2 \cdot \frac{3}{4}} \cdot a^{-\frac{1}{4} \cdot \frac{3}{4}}$

2. Simplify the exponents:

   • For $b^2$, the new exponent is: $2 \cdot \frac{3}{4} = \frac{6}{4} = \frac{3}{2}$
   • For $a^{-\frac{1}{4}}$, the new exponent is: $-\frac{1}{4} \cdot \frac{3}{4} = -\frac{3}{16}$

   Now the expression becomes: $b^{\frac{3}{2}} \cdot a^{-\frac{3}{16}}$

3. Convert negative exponents: To remove negative exponents, recall that $x^{-n} = \frac{1}{x^n}$. Thus, we move $a^{-\frac{3}{16}}$ to the denominator:

   $\frac{b^{\frac{3}{2}}}{a^{\frac{3}{16}}}$

Thus, the simplified expression is:

$\frac{b^{\frac{3}{2}}}{a^{\frac{3}{16}}}$

Would you like further details or clarifications?

Here are some related questions you could explore:

1. What is the general rule for simplifying expressions with fractional exponents?
2. How do you simplify expressions involving negative exponents?
3. How do you handle mixed bases in an expression like this one?
4. What are some common mistakes when simplifying exponentials?
5. How does exponentiation behave under multiplication of terms?

Tip: When simplifying expressions with multiple variables and exponents, always apply the power rules carefully and keep track of signs to avoid mistakes.