To evaluate \( 6^{\frac{3}{2}} \), we can break it down into two parts.
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First, we handle the exponentiation: \[ 6^{\frac{3}{2}} = (6^3)^{\frac{1}{2}} \]
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Now, calculate \( 6^3 \): \[ 6^3 = 216 \]
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Now, take the square root: \[ (6^3)^{\frac{1}{2}} = \sqrt{216} \]
Next, we can simplify \( \sqrt{216} \): \[ 216 = 36 \times 6 \] Thus, \[ \sqrt{216} = \sqrt{36 \times 6} = \sqrt{36} \cdot \sqrt{6} = 6\sqrt{6} \]
Now we can compare this result with the answer options:
A. \( \sqrt{18} \)
B. \( 3\sqrt{6} \)
C. \( 6\sqrt{6} \)
D. \( \sqrt[3]{12} \)
E. \( \sqrt[3]{36} \)
The correct equivalent of \( 6^{\frac{3}{2}} \) is: C. \( 6\sqrt{6} \).
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