Let's fully simplify the expression:

$\frac{\sqrt[3]{64x^5y^4}}{7x^2}$

Step 1: Simplify the cube root

We start by simplifying the cube root $\sqrt[3]{64x^5y^4}$:

• $\sqrt[3]{64} = \sqrt[3]{4^3} = 4$
• $\sqrt[3]{x^5} = x^{5/3}$
• $\sqrt[3]{y^4} = y^{4/3}$

Thus, the cube root simplifies to:

$\sqrt[3]{64x^5y^4} = 4x^{5/3}y^{4/3}$

Step 2: Combine the expression

Now, we have:

$\frac{4x^{5/3}y^{4/3}}{7x^2}$

Step 3: Simplify the powers of $x$

Using the rule $\frac{x^a}{x^b} = x^{a-b}$, we simplify the powers of $x$:

$\frac{x^{5/3}}{x^2} = x^{5/3 - 2} = x^{5/3 - 6/3} = x^{-1/3}$

Step 4: Write the final simplified expression

After simplifying, the final expression becomes:

$\frac{4y^{4/3}}{7x^{1/3}}$

This is the fully simplified version of the expression.

Do you want more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would you handle cube roots of negative numbers in similar expressions?
2. What is the general rule for simplifying fractional exponents?
3. How do you simplify expressions involving higher roots, like the fourth or fifth root?
4. How do you handle the combination of multiple variables within radical expressions?
5. How do you simplify complex fractions when both numerator and denominator involve radicals?

Tip: Always simplify powers and radicals separately before dealing with any fractions in an expression!