Question
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2. \( -\sqrt[3]{81} \)

Ask by Rowe Davison. in South Africa
Feb 03,2025

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Tutor-Verified Answer

Answer

\[ -\sqrt[3]{81} = -3\sqrt[3]{3} \]

Solution

To evaluate \( -\sqrt[3]{81} \), we can simplify it as follows: 1. **Express 81 as a Power of 3:** \[ 81 = 3^4 \] 2. **Apply the Cube Root:** \[ \sqrt[3]{81} = \sqrt[3]{3^4} = 3^{4/3} \] This can be broken down further: \[ 3^{4/3} = 3^{1 + 1/3} = 3 \cdot 3^{1/3} = 3\sqrt[3]{3} \] 3. **Include the Negative Sign:** \[ -\sqrt[3]{81} = -3\sqrt[3]{3} \] **Final Answer:** \[ -\sqrt[3]{81} = -3\,\sqrt[3]{\,3\,} \]

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Extra Insights

Did you know that the cube root of a number is asking the question: which number multiplied by itself three times gives us that number? For \( -\sqrt[3]{81} \), we can break it down to find that \( \sqrt[3]{81} = 4.326 \) approximately, so when we take the negative, we get about \( -4.326 \). It's like asking for the 'bad boy' version of that number! Now, let’s get practical! Negative cube roots can show up in various real-world applications, particularly in physics and engineering. For instance, when calculating forces in opposing directions, understanding how negatives operate can be crucial. So next time you face a negative cube root, remember, it’s just another twist in the mathematical adventure!

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