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

Answer 1

To match the expressions that are equivalent, we can simplify each expression separately.

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

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

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

4. **\( 2^{-2} \cdot 2^{-3} \)**
- Using the property of exponents \( a^m \cdot a^n = a^{m+n} \):
- \( 2^{-2} \cdot 2^{-3} = 2^{-2 + (-3)} = 2^{-5} \)

5. **\( 2^8 \cdot 2^{-3} \)**
- Again using the property of exponents \( a^m \cdot a^n = a^{m+n} \):
- \( 2^8 \cdot 2^{-3} = 2^{8 + (-3)} = 2^{5} \)

Now, we've simplified all expressions:
1. \( \frac{2^2}{2^3} = 2^{-1} \)
2. \( (2^2)^3 = 2^6 \)
3. \( \frac{2^3}{2^2} = 2^1 = 2 \)
4. \( 2^{-2} \cdot 2^{-3} = 2^{-5} \)
5. \( 2^8 \cdot 2^{-3} = 2^5 \)

Now, let's summarize the results:
- \( 2^{-1} \) (from expression 1)
- \( 2^6 \) (from expression 2)
- \( 2^1 \) or \( 2 \) (from expression 3)
- \( 2^{-5} \) (from expression 4)
- \( 2^5 \) (from expression 5)

None of these expressions yield matching equivalent results, but the equivalent pairs based on the simplified forms can be summarized as:
- **No matching equivalent expressions** were found among the given options.