What is the following product? \( \sqrt[3]{24} \cdot \sqrt[3]{45} \)

To simplify \( \sqrt[3]{24} \cdot \sqrt[3]{45} \), you can combine the cube roots: \[ \sqrt[3]{24 \cdot 45} = \sqrt[3]{1080}. \] Next, you can find the prime factorization of \( 1080 \): \( 1080 = 2^3 \cdot 3^3 \cdot 5 \). Hence, \[ \sqrt[3]{1080} = \sqrt[3]{(2^3) \cdot (3^3) \cdot 5} = 2 \cdot 3 \cdot \sqrt[3]{5} = 6\sqrt[3]{5}. \] So, \( \sqrt[3]{24} \cdot \sqrt[3]{45} = 6\sqrt[3]{5} \). Additionally, when dealing with cube roots like these, it's important to remember that they can often simplify dramatically if you break them down into their prime factor components. It’s a fun little puzzle that shows how number properties can work in your favor! Finally, for those who love visual aids, you might enjoy graphs that show the behavior of cube roots. They can help visualize how the numbers interact as you increase or decrease their values! Plus, there are vibrant mathematical communities online that delve into similar problems – it’s like a treasure hunt for number enthusiasts!