To simplify the expression \( \sqrt[3]{500a^4b^6} \), we will first break it down.
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Factor the constants:
- \( 500 = 5^3 \cdot 2 \) (since \( 500 = 125 \cdot 4 = 5^3 \times 2^2 \)).
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Break down the variables:
- \( a^4 = a^{3+1} = a^3 \cdot a^1 \).
- \( b^6 = (b^2)^3 \).
Now, we can rewrite the expression:
\[ \sqrt[3]{500a^4b^6} = \sqrt[3]{5^3 \cdot 2 \cdot a^3 \cdot a \cdot (b^2)^3} \]
Now apply the properties of cube roots:
\[ \sqrt[3]{5^3} = 5, \quad \sqrt[3]{a^3} = a, \quad \sqrt[3]{(b^2)^3} = b^2 \]
So we can simplify the expression:
\[ \sqrt[3]{500a^4b^6} = 5b^2a \sqrt[3]{2a} \]
Thus, the fully simplified expression is:
\[ 5ab^2 \sqrt[3]{2a} \]
Now, looking at your given choices, the closest simplified form is:
5ab² 4a−−√3 (assuming this means 5ab² 2√3).
However, it seems your multiple-choice options may not directly reflect the result I calculated or could contain formatting issues. Please ensure to double-check your options. If "4a" means "2a" in the expression context, that would validate the answer. Otherwise, please specify differently.
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