The expression given is:

$\frac{3yx^4}{6x^3}$

To simplify this and express the answer with only positive exponents, let’s break down each component:

Step 1: Simplify the Coefficients

The coefficients $3$ and $6$ simplify as follows: $\frac{3}{6} = \frac{1}{2}$

Step 2: Simplify the $x$ Terms Using Exponent Rules

The expression for $x$ in the numerator and denominator is $x^4$ and $x^3$, respectively. Using the rule $\frac{a^m}{a^n} = a^{m-n}$, we get: $x^{4 - 3} = x^1 = x$

Step 3: Combine the Results

After simplification, the expression becomes: $\frac{1}{2} \cdot y \cdot x = \frac{yx}{2}$

Final Answer

$\frac{yx}{2}$

Would you like a detailed explanation on any part of this simplification?

Here are 5 related questions to expand your understanding:

1. How would you simplify $\frac{4x^5 y^2}{8x^3 y}$?
2. What happens when there is a negative exponent in the denominator?
3. How do you handle variables in the numerator and denominator if they contain addition or subtraction?
4. Can you simplify an expression like $\frac{5a^3b}{10a^2b^2}$?
5. What if the exponents are zero in similar expressions?

Tip: Always reduce coefficients and simplify each variable separately when working with fractions involving exponents.