Let's simplify the expression you provided, which is the following:

[ \frac{\sqrt[3]{270x^{20}}}{\sqrt[3]{5x}} ]

Step 1: Apply the cube root to both terms

We can first simplify the expression by combining the cube roots into one:

[ \frac{\sqrt[3]{270x^{20}}}{\sqrt[3]{5x}} = \sqrt[3]{\frac{270x^{20}}{5x}} ]

Step 2: Simplify the fraction inside the cube root

Now, simplify the fraction:

$\frac{270x^{20}}{5x} = \frac{270}{5} \cdot \frac{x^{20}}{x} = 54x^{19}$

Step 3: Apply the cube root

Now, we take the cube root of the simplified expression:

$\sqrt[3]{54x^{19}}$

To break this down:

• The cube root of 54 can be simplified to $\sqrt[3]{54} = 54^{1/3}$.
• The cube root of $x^{19}$ is $x^{19/3}$, which can be further split as $x^{6 + 1/3} = x^6 \cdot x^{1/3}$.

Thus, the simplified expression is:

[ \sqrt[3]{54} \cdot x^6 \cdot \sqrt[3]{x} ]

Final Simplified Expression:

[ \sqrt[3]{54} \cdot x^6 \cdot \sqrt[3]{x} ]

This is the most simplified form of the given expression.

Would you like more details on any of these steps or have any further questions?

Here are some related questions you might find useful:

1. How do you simplify higher roots like square roots or fourth roots?
2. How do you handle fractions inside cube roots?
3. What is the general formula for simplifying nth roots in algebra?
4. How can you simplify expressions involving exponents and roots together?
5. What happens when you multiply cube roots? How is it different from other roots?

Tip: When simplifying cube roots, always look for the largest perfect cube factor that can be taken out of the radicand. This makes simplification faster!