Bot GPT 4-o mini To simplify \( \sqrt[3]{-72} \), we first break down \(-72\) into its factors.
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Write \(-72\) as: \[ -72 = -1 \times 72 \] Next, we can express \(72\) in terms of its prime factors: \[ 72 = 8 \times 9 = 8 \times 3^2 = 2^3 \times 3^2 \] Therefore, we can write: \[ -72 = -1 \times 2^3 \times 3^2 \]
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Now we apply the cube root to each factor: \[ \sqrt[3]{-72} = \sqrt[3]{-1 \times 2^3 \times 3^2} \] We can separate the cube roots: \[ \sqrt[3]{-1} \times \sqrt[3]{2^3} \times \sqrt[3]{3^2} \]
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Calculate each term:
- \( \sqrt[3]{-1} = -1 \)
- \( \sqrt[3]{2^3} = 2 \)
- \( \sqrt[3]{3^2} \) cannot be simplified further and is left as \( \sqrt[3]{9} \).
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Putting it all together: \[ \sqrt[3]{-72} = -1 \times 2 \times \sqrt[3]{9} = -2 \sqrt[3]{9} \]
Thus, the simplified form of \( \sqrt[3]{-72} \) is: \[ \boxed{-2 \sqrt[3]{9}} \]
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