To simplify \( \sqrt[3]{40} \), we begin by finding the prime factorization of 40.

\[
40 = 2^3 \times 5
\]

Now, apply the cube root:

\[
\sqrt[3]{40} = \sqrt[3]{2^3 \times 5} = \sqrt[3]{2^3} \times \sqrt[3]{5}
\]

The cube root of \( 2^3 \) simplifies to 2:

\[
\sqrt[3]{40} = 2 \times \sqrt[3]{5} = 2\sqrt[3]{5}
\]

Thus, the simplified form of \( \sqrt[3]{40} \) is:

\[
2\sqrt[3]{5}
\]

Would you like further details on cube roots or any other related concepts?

Here are 5 relative questions to expand your understanding:
1. How do you simplify expressions with higher-order roots, like the 4th root?
2. What is the difference between cube roots and square roots in simplification?
3. How would you handle the cube root of a fraction like \( \sqrt[3]{\frac{27}{8}} \)?
4. Can you approximate cube roots of non-perfect cubes without simplification?
5. How are cube roots used in solving cubic equations?

**Tip:** Always look for perfect powers in a number's prime factorization when simplifying roots.

Learn how to simplify the cube root of 40 (³√40) using prime factorization. This step-by-step guide breaks down the process to achieve the simplified form, 2³√5, covering essential concepts in algebra, roots, and radicals. Suitable for students in Grades 8-10.

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