The given expression is:

$\left( y \cdot y^{1/3} \right)^{3/2}$

Step-by-Step Simplification:

1. Combine the Bases Inside the Parentheses: Using the property of exponents $a^m \cdot a^n = a^{m+n}$: $y \cdot y^{1/3} = y^{1 + 1/3}$

   Simplify the exponent $1 + 1/3$: $1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}$

   So, $y \cdot y^{1/3} = y^{4/3}$.

2. Apply the Outer Exponent: Now the expression becomes: $\left( y^{4/3} \right)^{3/2}$

   Use the property of exponents $(a^m)^n = a^{m \cdot n}$: $y^{\frac{4}{3} \cdot \frac{3}{2}}$

   Simplify the exponent multiplication: $\frac{4}{3} \cdot \frac{3}{2} = \frac{4 \cdot 3}{3 \cdot 2} = \frac{12}{6} = 2$

   So the expression simplifies to: $y^2$

Final Answer:

$y^2$

Would you like further clarification or more details?

Related Questions:

1. What are the properties of exponents used in this solution?
2. How do you simplify expressions with fractional exponents?
3. What does it mean to multiply exponents when raising a power to another power?
4. How do fractional exponents relate to roots (e.g., $y^{1/3}$)?
5. Can we generalize the solution for any similar expression?

Tip:

When simplifying exponents, always combine like terms step-by-step to avoid confusion, especially when dealing with fractional exponents.