Match the expressions that are equivalent.(5 points)

Put responses in the correct input to answer the question.
1. 2^3/2^2:
2. 2^−2 × 2^−3:
3. 2^8 × 2^−3:
4. (2^2)^3:
5. 2^2/2^3:

1 answer

Let's evaluate each expression to find the equivalent ones:

  1. \( \frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2 \)

  2. \( 2^{-2} \times 2^{-3} = 2^{-2 + (-3)} = 2^{-5} \)

  3. \( 2^8 \times 2^{-3} = 2^{8 + (-3)} = 2^{5} \)

  4. \( (2^2)^3 = 2^{2 \times 3} = 2^{6} \)

  5. \( \frac{2^2}{2^3} = 2^{2-3} = 2^{-1} = \frac{1}{2} \)

Now we can summarize the results:

  1. \( 2 \)
  2. \( 2^{-5} \)
  3. \( 2^{5} \)
  4. \( 2^{6} \)
  5. \( \frac{1}{2} \)

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!