We are given the expression:

$\frac{2x^{-\frac{7}{4}}}{4x^{\frac{4}{3}}}$

Let's simplify this step by step:

Step 1: Simplify the constants

We have $\frac{2}{4} = \frac{1}{2}$, so the expression becomes:

$\frac{1}{2} \cdot \frac{x^{-\frac{7}{4}}}{x^{\frac{4}{3}}}$

Step 2: Apply the quotient rule of exponents

The quotient rule of exponents states:

$\frac{a^m}{a^n} = a^{m-n}$

Using this rule for $x^{-\frac{7}{4}}$ and $x^{\frac{4}{3}}$, we subtract the exponents:

$x^{-\frac{7}{4} - \frac{4}{3}} = x^{-\frac{7}{4} - \frac{4}{3}}$

Step 3: Find the common denominator

To subtract the exponents, we need to express them with a common denominator. The denominators 4 and 3 have a least common denominator of 12, so we rewrite the exponents:

$-\frac{7}{4} = -\frac{21}{12}, \quad \frac{4}{3} = \frac{16}{12}$

Now, subtract:

$-\frac{21}{12} - \frac{16}{12} = -\frac{37}{12}$

So, the expression becomes:

$\frac{1}{2} \cdot x^{-\frac{37}{12}} = \frac{x^{-\frac{37}{12}}}{2}$

Final Answer:

$\frac{x^{-\frac{37}{12}}}{2}$

Would you like to explore the result in more depth, or ask any questions? Here are some related problems:

1. Simplify $\frac{3x^{-\frac{5}{3}}}{6x^{\frac{2}{3}}}$.
2. What is the result of $\frac{5x^{-2}}{10x^{\frac{5}{2}}}$?
3. How do you simplify $\frac{4x^{\frac{7}{3}}}{8x^{-\frac{4}{3}}}$?
4. Apply the exponent rules to $\frac{x^{-\frac{3}{4}}}{x^{\frac{7}{6}}}$.
5. Simplify $\frac{9x^{\frac{1}{2}}}{3x^{-\frac{5}{6}}}$.

Tip: When dealing with fractional exponents, always aim to rewrite them with a common denominator for easier subtraction or addition.