Question
Which expressions are equivalent to -64? Check TWO that apply.(2 points)
Responses
4−3
4 to the negative 3 power
−(14)−3
negative 1 fourth to the negative 3 power
28−2422
the fraction with numerator 2 to the 8th power minus 2 to the 4th power and denominator 2 squared
23⋅22⋅20
2 cubed times 2 squared times 2 to the 0 power
−2 − superscript2 ⋅ ( 2 superscript8 )
CHECK 2 THAT APPLY
Responses
4−3
4 to the negative 3 power
−(14)−3
negative 1 fourth to the negative 3 power
28−2422
the fraction with numerator 2 to the 8th power minus 2 to the 4th power and denominator 2 squared
23⋅22⋅20
2 cubed times 2 squared times 2 to the 0 power
−2 − superscript2 ⋅ ( 2 superscript8 )
CHECK 2 THAT APPLY
Answers
GPT-4o mini
Let's evaluate each expression to determine which are equivalent to -64.
1. **\(4^{-3}\)**:
\[
4^{-3} = \frac{1}{4^3} = \frac{1}{64}
\]
This is **not equivalent** to -64.
2. **\(- (1/4)^{-3}\)**:
\[
- (1/4)^{-3} = - (4^3) = -64
\]
This **is equivalent** to -64.
3. **\(\frac{2^8 - 2^4}{2^2}\)**:
First, let's calculate \(2^8 - 2^4\):
\[
2^8 = 256 \quad \text{and} \quad 2^4 = 16 \quad \Rightarrow \quad 2^8 - 2^4 = 256 - 16 = 240
\]
Now dividing by \(2^2\):
\[
\frac{240}{2^2} = \frac{240}{4} = 60
\]
This is **not equivalent** to -64.
4. **\(2^3 \cdot 2^2 \cdot 2^0\)**:
\[
2^3 \cdot 2^2 \cdot 2^0 = 2^{3+2+0} = 2^5 = 32
\]
This is **not equivalent** to -64.
5. **\(-2^2 \cdot (2^8)\)**:
\[
-2^2 \cdot 2^8 = -4 \cdot 256 = -1024
\]
This is **not equivalent** to -64.
Based on the evaluations, the two expressions that are equivalent to -64 are:
- \(- (1/4)^{-3}\)
- None of the other expressions evaluate to -64.
Thus, the only correct response is \(- (1/4)^{-3}\). Please check if there might be an error in the list provided, as one option is needed to meet the selection criteria.
1. **\(4^{-3}\)**:
\[
4^{-3} = \frac{1}{4^3} = \frac{1}{64}
\]
This is **not equivalent** to -64.
2. **\(- (1/4)^{-3}\)**:
\[
- (1/4)^{-3} = - (4^3) = -64
\]
This **is equivalent** to -64.
3. **\(\frac{2^8 - 2^4}{2^2}\)**:
First, let's calculate \(2^8 - 2^4\):
\[
2^8 = 256 \quad \text{and} \quad 2^4 = 16 \quad \Rightarrow \quad 2^8 - 2^4 = 256 - 16 = 240
\]
Now dividing by \(2^2\):
\[
\frac{240}{2^2} = \frac{240}{4} = 60
\]
This is **not equivalent** to -64.
4. **\(2^3 \cdot 2^2 \cdot 2^0\)**:
\[
2^3 \cdot 2^2 \cdot 2^0 = 2^{3+2+0} = 2^5 = 32
\]
This is **not equivalent** to -64.
5. **\(-2^2 \cdot (2^8)\)**:
\[
-2^2 \cdot 2^8 = -4 \cdot 256 = -1024
\]
This is **not equivalent** to -64.
Based on the evaluations, the two expressions that are equivalent to -64 are:
- \(- (1/4)^{-3}\)
- None of the other expressions evaluate to -64.
Thus, the only correct response is \(- (1/4)^{-3}\). Please check if there might be an error in the list provided, as one option is needed to meet the selection criteria.