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.

Learn how to simplify the exponential expression [2(x)^-7/4] / [4(x)^4/3] using the quotient rule of exponents. This step-by-step guide covers key concepts like simplifying rational expressions and applying exponent rules. Suitable for students in Grades 9-12.

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