Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 2\, \sqrt[3]{10} \)
Solution
To simplify \( \sqrt[3]{80} \), follow these steps:
1. **Factorize 80 into its prime factors:**
\[
80 = 2^4 \times 5
\]
2. **Identify any perfect cubes within the factors:**
\[
2^4 = 2^3 \times 2
\]
Here, \( 2^3 = 8 \) is a perfect cube.
3. **Rewrite the cube root using the perfect cube:**
\[
\sqrt[3]{80} = \sqrt[3]{2^3 \times 2 \times 5} = \sqrt[3]{2^3} \times \sqrt[3]{2 \times 5}
\]
4. **Simplify the expression:**
\[
\sqrt[3]{2^3} = 2 \quad \text{and} \quad \sqrt[3]{2 \times 5} = \sqrt[3]{10}
\]
5. **Combine the simplified parts:**
\[
\sqrt[3]{80} = 2 \sqrt[3]{10}
\]
**Final Answer:**
\[
2\, \sqrt[3]{10}
\]
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Simplify this solution The Deep Dive
To simplify \( \sqrt[3]{80} \), we can start by breaking down the number 80 into its prime factors. The prime factorization of 80 is \( 2^4 \times 5 \). Now, applying the property of cube roots, we can separate the factors: \[ \sqrt[3]{80} = \sqrt[3]{2^4 \times 5} = \sqrt[3]{2^3 \times 2 \times 5} = \sqrt[3]{2^3} \times \sqrt[3]{2 \times 5} \] Since \( \sqrt[3]{2^3} = 2 \), we have: \[ \sqrt[3]{80} = 2 \times \sqrt[3]{10} \] Thus, the simplified form is \( 2\sqrt[3]{10} \).
