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To simplify \( 4 \frac{1}{2} \):
To convert the mixed number to an improper fraction: \[ 4 \frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2}. \] So, \( 4 \frac{1}{2} = \frac{9}{2} \).
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To rewrite \( \sqrt{20}^5 \) using a rational number as an exponent:
The square root can be expressed as an exponent of \( \frac{1}{2} \): \[ \sqrt{20} = 20^{\frac{1}{2}}. \] Therefore, \[ \sqrt{20}^5 = (20^{\frac{1}{2}})^5 = 20^{\frac{5}{2}}. \] The rational number that could be used as an exponent is \( \frac{5}{2} \).
In summary:
- \( 4 \frac{1}{2} = \frac{9}{2} \)
- The rational number that could be used as an exponent is \( \frac{5}{2} \).
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