To simplify the given expression:

$\left(\frac{5x^{\frac{2}{3}}}{y^{\frac{1}{2}}}\right)(2x^2y^{-3})$

Step 1: Combine constants and separate variables

The expression can be written as:

$(5 \cdot 2) \cdot \left(x^{\frac{2}{3}} \cdot x^2\right) \cdot \left(\frac{1}{y^{\frac{1}{2}}} \cdot y^{-3}\right)$

This simplifies the constants and separates the powers of $x$ and $y$.

Step 2: Simplify the constants

$5 \cdot 2 = 10$

Step 3: Simplify the powers of $x$

Using the rule of exponents $a^m \cdot a^n = a^{m+n}$:

$x^{\frac{2}{3}} \cdot x^2 = x^{\frac{2}{3} + 2} = x^{\frac{2}{3} + \frac{6}{3}} = x^{\frac{8}{3}}$

Step 4: Simplify the powers of $y$

Using the same rule of exponents:

$\frac{1}{y^{\frac{1}{2}}} \cdot y^{-3} = y^{-\frac{1}{2} - 3} = y^{-\frac{1}{2} - \frac{6}{2}} = y^{-\frac{7}{2}}$

Step 5: Combine the results

Now, putting everything together:

$10 \cdot x^{\frac{8}{3}} \cdot y^{-\frac{7}{2}}$

This can be rewritten for clarity as:

$\boxed{\frac{10x^{\frac{8}{3}}}{y^{\frac{7}{2}}}}$

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Would you like me to break down any step further or solve similar problems?

Here are 5 related questions you might find helpful:

1. How do you handle fractional exponents when dividing powers?
2. Can you explain the rules for combining exponents in general?
3. What is the significance of negative exponents in simplifications?
4. How would you solve the expression if the base values for $x$ and $y$ were given?
5. Can you explain how fractional powers relate to roots (e.g., $x^{1/2}$)?

Tip: Always double-check your addition or subtraction of fractional exponents by ensuring a common denominator is used properly.