We are tasked with simplifying the expression:

$\sqrt[4]{256u^{12}}$

Step 1: Simplify the constant

Express 256 as a power of 2:

$256 = 2^8$

So the expression becomes:

$\sqrt[4]{2^8 \cdot u^{12}}$

Step 2: Apply the property of radicals

Recall that:

$\sqrt[n]{a^m} = a^{\frac{m}{n}}$

Applying this property:

$\sqrt[4]{2^8} = 2^{\frac{8}{4}} = 2^2 = 4$

$\sqrt[4]{u^{12}} = u^{\frac{12}{4}} = u^3$

Step 3: Final simplified expression

$4u^3$

Final Answer:

$4u^3$

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to explore:

1. How do you simplify higher-order roots involving negative exponents?
2. What is the general rule for simplifying $\sqrt[n]{a^m}$?
3. How would the answer change if $u$ were a negative real number?
4. Can you simplify $\sqrt[3]{81x^9}$?
5. How do you rationalize the denominator in expressions involving roots?

Tip: Always factor constants into their prime bases when simplifying roots—it makes the process much easier!