To estimate \( 32^{0.8} \), we can use the fact that \( 0.8 \) is equivalent to \( 4/5 \). Thus, we can rewrite \( 32^{0.8} \) as \( 32^{4/5} \).
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Calculate \( 32^1 \):
\[ 32^1 = 32 \] -
Calculate \( 32^{0} \) for verification:
\[ 32^0 = 1 \]
We know that \( 32^{0.8} \) lies between \( 32^0 \) and \( 32^1 \). Since \( 0.8 \) is closer to \( 1 \), we expect \( 32^{0.8} \) to be closer to \( 32 \) than to \( 1 \).
For a more refined estimate, we can calculate \( 32^{0.8} \) using logarithms or a calculator, which gives approximately \( 22.63 \).
Among the given options:
- 22.4
- 25.6
- 32
- 16
The best estimate for \( 32^{0.8} \) is 22.4.