Let's evaluate each expression to find the equivalent ones:
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\( \frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2 \)
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\( 2^{-2} \times 2^{-3} = 2^{-2 + (-3)} = 2^{-5} \)
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\( 2^8 \times 2^{-3} = 2^{8 + (-3)} = 2^{5} \)
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\( (2^2)^3 = 2^{2 \times 3} = 2^{6} \)
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\( \frac{2^2}{2^3} = 2^{2-3} = 2^{-1} = \frac{1}{2} \)
Now we can summarize the results:
- \( 2 \)
- \( 2^{-5} \)
- \( 2^{5} \)
- \( 2^{6} \)
- \( \frac{1}{2} \)
The equivalent expressions are:
- Expression 1 (\(2^3/2^2\)) is equivalent to \(2\).
- Expression 3 (\(2^8 \times 2^{-3}\)) is equivalent to \(2^5\).
- Expression 4 (\((2^2)^3\)) is equivalent to \(2^6\).
- Expression 5 (\(2^2/2^3\)) is equivalent to \(\frac{1}{2}\).
- Expression 2 (\(2^{-2} \times 2^{-3}\)) is equivalent to \(2^{-5}\).
So we can group them as follows:
- \(2^3/2^2 \equiv 2 \)
- \(2^8 × 2^{-3} \equiv 2^5\)
- \((2^2)^3 \equiv 2^6\)
- \( 2^2/2^3 \equiv \frac{1}{2} \)
- \( 2^{-2} × 2^{-3} \equiv 2^{-5} \)
This provides a good framing for grouping the outputs. However, since no specific equivalent expressions were required, there's no explicit matching required in a traditional sense—each expression has its unique outcome when simplified. If you need a direct answer based on equivalency or matching, please specify how you prefer it formatted!
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