Simplify the expression. $$ x^{-\frac{3}{2}} x^{\frac{1}{3}} x^{\frac{3}{4}} $$ Write your answer using only positive exponents. Assume that all variables are positive real numbers.

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We start with the expression:
$$
x^{-\frac{3}{2}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{3}{4}}.
$$
Since the bases are the same, we add the exponents:
$$
-\frac{3}{2} + \frac{1}{3} + \frac{3}{4}.
$$
To add these fractions, we use the common denominator $$12$$:
$$
-\frac{3}{2} = -\frac{18}{12}, \quad \frac{1}{3} = \frac{4}{12}, \quad \frac{3}{4} = \frac{9}{12}.
$$
Now we add the numerators:
$$
-\frac{18}{12} + \frac{4}{12} + \frac{9}{12} = \frac{-18 + 4 + 9}{12} = \frac{-5}{12}.
$$
Thus, the expression simplifies to:
$$
x^{-\frac{5}{12}}.
$$
Since we require only positive exponents, we write:
$$
\frac{1}{x^{\frac{5}{12}}}.
$$
The expression simplifies to $$ \frac{1}{x^{\frac{5}{12}}} $$.

Simplify the expression.

Write your answer using only positive exponents.
Assume that all variables are positive real numbers.

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Simplify the expression. Write your answer using only positive exponents. Assume that all variables are positive real numbers.