We are asked to simplify the expression:

$3 \cdot \sqrt[3]{x} \cdot x^{7/3} \cdot y^{5/2} \cdot \sqrt{y^3}$

Step 1: Convert radicals into fractional exponents

• The cubed root of $x$ can be written as $x^{1/3}$.
• The square root of $y^3$ is $y^{3/2}$.

Thus, the expression becomes:

$3 \cdot x^{1/3} \cdot x^{7/3} \cdot y^{5/2} \cdot y^{3/2}$

Step 2: Combine terms with the same base

For $x$:

Use the rule $x^a \cdot x^b = x^{a+b}$.

$x^{1/3} \cdot x^{7/3} = x^{(1/3 + 7/3)} = x^{8/3}$

For $y$:

$y^{5/2} \cdot y^{3/2} = y^{(5/2 + 3/2)} = y^{8/2} = y^4$

Step 3: Write the simplified expression

The expression is now:

$3 \cdot x^{8/3} \cdot y^4$

Step 4: Convert back to radical form

Now, rewrite the expression in terms of radicals:

• $x^{8/3}$ can be written as $\sqrt[3]{x^8}$.
• $y^4$ remains as is, since it's already in a simplified form.

Thus, the final simplified expression in radical form is:

$3 \cdot \sqrt[3]{x^8} \cdot y^4$

Would you like further details or clarifications on this process?

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Here are 5 related questions to deepen your understanding:

1. How do you simplify expressions with different radicals like cube roots and square roots together?
2. How would the expression change if $x^7/3$ were $x^{2/3}$?
3. What happens if one of the exponents is negative—how would that affect the simplification?
4. Can you simplify $y^{5/2} \cdot y^{-1/2}$ and express it in radical form?
5. How can you generalize this process for any combination of radicals and exponents?

Tip: When working with exponents and radicals, always try converting everything into fractional exponents first; it makes combining and simplifying terms much easier.