To simplify the expression \( \sqrt[3]{500a^4b^6} \), let's break it down step-by-step.
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Factor the constant:
- We start with \( 500 \). The prime factorization of \( 500 \) is \( 5^3 \times 2 \).
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Express the variables:
- The expression \( a^4 \) can be written as \( (a^3) \cdot a \).
- The expression \( b^6 \) can be written as \( (b^6) \) or \( (b^3)^2 \).
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Rewrite the expression using these factors:
- Therefore, we have: \[ \sqrt[3]{500a^4b^6} = \sqrt[3]{5^3 \cdot 2 \cdot (a^3 \cdot a) \cdot (b^3)^2} \]
- This simplifies to: \[ \sqrt[3]{5^3} \cdot \sqrt[3]{2} \cdot \sqrt[3]{a^3} \cdot \sqrt[3]{a} \cdot \sqrt[3]{b^6} \]
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Evaluate the cube roots:
- \( \sqrt[3]{5^3} = 5 \)
- \( \sqrt[3]{a^3} = a \)
- \( \sqrt[3]{b^6} = b^2 \)
- Therefore, the expression becomes: \[ 5 \cdot a \cdot b^2 \cdot \sqrt[3]{2 \cdot a} \]
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Combine all parts:
- Bringing it together, we have: \[ 5ab^2 \sqrt[3]{2a} \]
Thus the simplified form of \( \sqrt[3]{500a^4b^6} \) is: \[ 5ab^2 \sqrt[3]{2a} \]
None of the given responses perfectly fits the simplification, but the closest would be \( 5ab^2 \sqrt[3]{2a} \). Please check if there are similar terms in the options provided or the original prompt.