Question

Irrational Exponents Quick Check
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Question
Which of the following is the best estimate for 320.8
?(1 point)
Responses

22.4
22.4

25.6
25.6

32
32

16
16
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Answer 1

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 \( 32^{4/5} \).

1. Calculate \( 32^1 \):
\[
32^1 = 32
\]

2. 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**.