Simplify the expression 3 cubed root x times x^7/3 times y^5/2 times square root y^3 
 and rewrite it in radical form.

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.

Simplify the expression 3 cubed root x times x^7/3 times y^5/2 times square root y^3 and rewrite it in radical form.

Learn how to simplify the expression 3 cubed root x times x^7/3 times y^5/2 times square root y^3 into radical form. This step-by-step guide covers algebraic rules for exponents and radicals. Ideal for students in Grades 9-12.

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