Question

Match the expressions that are equivalent.
2 to the second power divided by 2 to the third power
(2 to the second power) to the third power
2 to the third power divided by 2 to the second power
2 to the negative second power times 2 to the negative third power
2 to the eighth power times 2 to the negative third power

32
1/2
64
2
1/32
1/64

Answer 1

Let's simplify each expression step-by-step and then match them to their equivalent values.

1. **\( \frac{2^2}{2^3} \)**:
- Using the properties of exponents: \( \frac{a^m}{a^n} = a^{m-n} \)
- So, \( \frac{2^2}{2^3} = 2^{2-3} = 2^{-1} = \frac{1}{2} \)

2. **\( (2^2)^3 \)**:
- Using the power of a power property: \( (a^m)^n = a^{m \cdot n} \)
- So, \( (2^2)^3 = 2^{2 \cdot 3} = 2^6 = 64 \)

3. **\( \frac{2^3}{2^2} \)**:
- Again using the division property of exponents: \( \frac{a^m}{a^n} = a^{m-n} \)
- So, \( \frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2 \)

4. **\( 2^{-2} \times 2^{-3} \)**:
- Using the product of powers property: \( a^m \times a^n = a^{m+n} \)
- So, \( 2^{-2} \times 2^{-3} = 2^{-2-3} = 2^{-5} = \frac{1}{2^5} = \frac{1}{32} \)

5. **\( 2^8 \times 2^{-3} \)**:
- Using the product of powers property again:
- So, \( 2^8 \times 2^{-3} = 2^{8-3} = 2^5 = 32 \)

Now, matching the results:

1. \( \frac{2^2}{2^3} = \frac{1}{2} \)
2. \( (2^2)^3 = 64 \)
3. \( \frac{2^3}{2^2} = 2 \)
4. \( 2^{-2} \times 2^{-3} = \frac{1}{32} \)
5. \( 2^8 \times 2^{-3} = 32 \)

Final equivalent matches:
- \( \frac{2^2}{2^3} \) : \( \frac{1}{2} \)
- \( (2^2)^3 \) : \( 64 \)
- \( \frac{2^3}{2^2} \) : \( 2 \)
- \( 2^{-2} \times 2^{-3} \) : \( \frac{1}{32} \)
- \( 2^8 \times 2^{-3} \) : \( 32 \)

The equivalent values for the original expressions are:
1. \(\frac{2^2}{2^3} = \frac{1}{2}\)
2. \((2^2)^3 = 64\)
3. \(\frac{2^3}{2^2} = 2\)
4. \(2^{-2} \times 2^{-3} = \frac{1}{32}\)
5. \(2^{8} \times 2^{-3} = 32\)

So the final groupings become:
- \( \frac{2^2}{2^3} \) → \( \frac{1}{2} \)
- \( (2^2)^3 \) → \( 64 \)
- \( \frac{2^3}{2^2} \) → \( 2 \)
- \( 2^{-2} \times 2^{-3} \) → \( \frac{1}{32} \)
- \( 2^{8} \times 2^{-3} \) → \( 32 \)