Question

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

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

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

4^−3
4 to the negative 3 power

−(1/4)^−3
negative 1 fourth to the negative 3 power

2^8−2^4/2^2

Answer 1

To determine which expressions are equivalent to -64, let's evaluate each expression step by step.

1. **\( 2^3 \cdot 2^2 \cdot 2^0 \)**:
\[
2^3 \cdot 2^2 \cdot 2^0 = 2^{3+2+0} = 2^5 = 32
\]
This is not equivalent to -64.

2. **\( -2^{-2} \cdot (2^8) \)**:
\[
-2^{-2} \cdot 2^8 = -\frac{1}{2^2} \cdot 2^8 = -\frac{1}{4} \cdot 256 = -64
\]
This expression is equivalent to -64.

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

4. **\( -\left(\frac{1}{4}\right)^{-3} \)**:
\[
-\left(\frac{1}{4}\right)^{-3} = -4^3 = -64
\]
This expression is equivalent to -64.

5. **\( 2^{8} - \frac{2^{4}}{2^{2}} \)**:
\[
2^{8} - \frac{2^{4}}{2^{2}} = 2^8 - 2^{4-2} = 2^8 - 2^2 = 256 - 4 = 252
\]
This is not equivalent to -64.

Therefore, the two expressions that are equivalent to -64 are:

- **\( -2^{-2} \cdot (2^8) \)**
- **\( -\left(\frac{1}{4}\right)^{-3} \)**