Question

Which expressions are equivalent to -64? Check TWO that apply.(2 points)
Responses

−2−^2⋅(2^8)
negative 2 to the negative 2 power times open paren 2 to the 8th power close paren

2^3⋅2^2⋅2^0
2 cubed times 2 squared times 2 to the 0 power

4−^3
4 to the negative 3 power

2^8−2^4/2^2
the fraction with numerator 2 to the 8th power minus 2 to the 4th power and denominator 2 squared

−(1/4)−^3

Answer 1

To determine which expressions are equivalent to -64, let's evaluate each option and see if it simplifies to -64.

1. **−2^(-2)⋅(2^8)**:
- This can be simplified as:
\[
-2^{-2} \cdot 2^8 = -\frac{1}{2^2} \cdot 2^8 = -\frac{1}{4} \cdot 256 = -64.
\]

2. **2^3⋅2^2⋅2^0**:
- This can be simplified using the property of exponents (adding the exponents):
\[
2^3 \cdot 2^2 \cdot 2^0 = 2^{3+2+0} = 2^5 = 32.
\]
This is not equivalent to -64.

3. **4^(-3)**:
- This can be rewritten as:
\[
4^{-3} = \frac{1}{4^3} = \frac{1}{64}.
\]
This is not equivalent to -64.

4. **2^8 - 2^4 / 2^2**:
- According to the order of operations, we compute:
\[
2^8 - \frac{2^4}{2^2} = 2^8 - 2^{4-2} = 2^8 - 2^2 = 256 - 4 = 252.
\]
This is not equivalent to -64.

5. **−(1/4)−^3**:
- This can be simplified as follows:
\[
-\left(\frac{1}{4}\right)^{-3} = -\left(\frac{4}{1}\right)^3 = -64.
\]

Based on the analysis above, the expressions equivalent to -64 are:

- **−2^(-2)⋅(2^8)**
- **−(1/4)−^3**

So, the correct responses are the first and the last expression.