The expression provided is:

[ \sqrt[3]{6xy^3} \cdot \sqrt[3]{18x^4y} ]

Let me simplify it step by step.

Step 1: Combine the cube roots

Using the property of radicals: [ \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} ] We can combine the two cube roots into a single one: $\sqrt[3]{6xy^3 \cdot 18x^4y}$

Step 2: Multiply the terms inside the cube root

Multiply the coefficients, variables, and their powers: $6 \cdot 18 = 108, \quad x \cdot x^4 = x^5, \quad y^3 \cdot y = y^4$ Thus: [ \sqrt[3]{6xy^3 \cdot 18x^4y} = \sqrt[3]{108x^5y^4} ]

Step 3: Simplify the cube root

Now, simplify $\sqrt[3]{108x^5y^4}$:

• Factor $108$ into cubes and leftover terms: $108 = 27 \cdot 4, \quad \text{where } 27 = 3^3 \text{ (a perfect cube)}.$ So: [ \sqrt[3]{108} = \sqrt[3]{27 \cdot 4} = \sqrt[3]{27} \cdot \sqrt[3]{4} = 3\sqrt[3]{4}. ]

• For $x^5$, rewrite it as $x^3 \cdot x^2$: [ \sqrt[3]{x^5} = \sqrt[3]{x^3 \cdot x^2} = \sqrt[3]{x^3} \cdot \sqrt[3]{x^2} = x \cdot \sqrt[3]{x^2}. ]

• For $y^4$, rewrite it as $y^3 \cdot y$: [ \sqrt[3]{y^4} = \sqrt[3]{y^3 \cdot y} = \sqrt[3]{y^3} \cdot \sqrt[3]{y} = y \cdot \sqrt[3]{y}. ]

Step 4: Combine the results

Putting everything together: [ \sqrt[3]{108x^5y^4} = 3xy \cdot \sqrt[3]{4x^2y}. ]

Thus, the simplified expression is: $3xy \cdot \sqrt[3]{4x^2y}.$

Final Answer:

$3xy \cdot \sqrt[3]{4x^2y}.$

Let me know if you would like me to explain any specific step in more detail! 😊

Related Questions:

1. What are the properties of cube roots and how can they simplify expressions?
2. How do you factor numbers to identify perfect cubes?
3. Can cube roots of negative numbers be simplified using the same rules?
4. What are the differences between square roots and cube roots in algebraic simplification?
5. How can we visualize cube roots geometrically or conceptually?

Tip:

Always look for perfect cubes in numerical and variable factors when simplifying cube roots to make the process faster and cleaner.