Let's simplify the given expression:

$\frac{ \left( x^{7/4} y^{-1/2} \cdot y \cdot x^{2/3} \right)^{3/4} }{ x^{1/2} y^{3/2} }$

Step 1: Simplify the terms inside the parentheses

We start by combining the terms inside the parentheses $(x^{7/4} y^{-1/2} \cdot y \cdot x^{2/3})$.

For $y^{-1/2} \cdot y$, apply the rule $y^a \cdot y^b = y^{a+b}$:

$y^{-1/2} \cdot y^1 = y^{(-1/2 + 1)} = y^{1/2}$

For $x^{7/4} \cdot x^{2/3}$, apply the rule $x^a \cdot x^b = x^{a+b}$. First, we need to express $2/3$ as a fraction with denominator 12, so:

$x^{7/4} \cdot x^{2/3} = x^{7/4 + 8/12} = x^{21/12 + 8/12} = x^{29/12}$

Thus, the expression inside the parentheses becomes:

$x^{29/12} y^{1/2}$

Step 2: Apply the exponent $3/4$ to each term

Now we raise each term inside the parentheses to the power $3/4$:

$\left( x^{29/12} y^{1/2} \right)^{3/4} = x^{(29/12) \cdot (3/4)} \cdot y^{(1/2) \cdot (3/4)}$

Simplifying the exponents:

$x^{(29/12) \cdot (3/4)} = x^{87/48} = x^{29/16}$ $y^{(1/2) \cdot (3/4)} = y^{3/8}$

Thus, we have:

$x^{29/16} y^{3/8}$

Step 3: Simplify the entire fraction

We now divide the expression $x^{29/16} y^{3/8}$ by $x^{1/2} y^{3/2}$:

$\frac{x^{29/16} y^{3/8}}{x^{1/2} y^{3/2}}$

Apply the rule $\frac{x^a}{x^b} = x^{a-b}$ to both the $x$-terms and the $y$-terms:

For the $x$-terms:

$x^{29/16} \cdot x^{-1/2} = x^{29/16 - 8/16} = x^{21/16}$

For the $y$-terms:

$y^{3/8} \cdot y^{-3/2} = y^{3/8 - 12/8} = y^{-9/8}$

Final Simplified Expression:

$x^{21/16} y^{-9/8}$

This is the simplified form of the given expression.

Would you like further details or have any questions?

Related Questions:

1. How do we combine exponents when multiplying terms with the same base?
2. What is the general rule for distributing exponents over products?
3. How can you simplify expressions with fractional exponents?
4. How do negative exponents affect the final expression?
5. Why do we need to express exponents with common denominators when adding them?

Tip: When dealing with exponents, it’s useful to break the problem down into smaller steps and use the properties of exponents methodically.