To simplify $\root(3)(320)$, we use the product property of radicals:

$\root(3)(a \cdot b) = \root(3)(a) \cdot \root(3)(b)$

Step 1: Prime factorize $320$

$320 = 2^6 \cdot 5$

Step 2: Group the factors into perfect cubes

• A perfect cube has powers divisible by 3. From $2^6$, we can write $2^6 = (2^3)^2$.
• The expression becomes: $320 = (2^3)^2 \cdot 5$

Step 3: Simplify the cube root

Using the product property of radicals: $\root(3)(320) = \root(3)((2^3)^2 \cdot 5) = \root(3)((2^3)^2) \cdot \root(3)(5)$

• Simplify $\root(3)((2^3)^2) = 2^2 = 4$.
• $\root(3)(5)$ remains as it is because 5 is not a perfect cube.

Final Answer:

$\root(3)(320) = 4 \cdot \root(3)(5)$

If you have questions, let me know! Here are 5 related problems to practice:

1. Simplify $\root(3)(540)$.
2. Simplify $\root(3)(108)$.
3. Simplify $\sqrt(72)$.
4. Simplify $\root(4)(512)$.
5. Simplify $\root(3)(8x^6)$.

Tip: Always check for the largest perfect power that can be factored out when simplifying radicals!