To simplify the expression \( \frac{12^5}{8} \cdot 12^{-1/2} \), let's break it down step-by-step.
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Rewrite the expression: \[ \frac{12^5}{8} \cdot 12^{-1/2} = \frac{12^5 \cdot 12^{-1/2}}{8} \]
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Combine the powers of 12: \[ 12^5 \cdot 12^{-1/2} = 12^{5 - 1/2} = 12^{5 - 0.5} = 12^{4.5} = 12^{\frac{9}{2}} \]
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Now, rewrite the whole expression: \[ \frac{12^{\frac{9}{2}}}{8} \]
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Recognize that \( 8 = 2^3 \) and consider \( 12 \): \[ 12 = 2^2 \cdot 3 \] Thus: \[ 12^{\frac{9}{2}} = (2^2 \cdot 3)^{\frac{9}{2}} = 2^{9} \cdot 3^{\frac{9}{2}} \]
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Substitute this into the expression: \[ \frac{2^{9} \cdot 3^{\frac{9}{2}}}{2^3} = 2^{9 - 3} \cdot 3^{\frac{9}{2}} = 2^{6} \cdot 3^{\frac{9}{2}} \]
Now, let's express this in the root form. The index of the root corresponds to the fractional exponents in the expression, specifically:
- The \( 3^{\frac{9}{2}} \) part can be rewritten using a square root: \[ 3^{\frac{9}{2}} = 3^{4.5} = 3^4 \cdot 3^{\frac{1}{2}} = 81 \cdot \sqrt{3} \] Here, the square root has an index of 2.
So, the expression \( 2^6 \cdot 3^{4.5} \) does not inherently require any roots; however, the \( \sqrt{3} \) we derived indicates that if we convert nature of the term into a root form (only considering the LAST part involving a square root) it involves an index of 2.
Therefore, the root form's index of the whole expression will be \(2\).
Final answer: The index of the root form of the expression is 2.
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