Question

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

23⋅22⋅20
2 cubed times 2 squared times 2 to the 0 power

28−2422
the fraction with numerator 2 to the 8th power minus 2 to the 4th power and denominator 2 squared

−2−2⋅(28)
negative 2 to the negative 2 power times open paren 2 to the 8th power close paren

−(14)−3
negative 1 fourth to the negative 3 power

4−3
4 to the negative 3 power

Answer 1

To determine which of the expressions are equivalent to -64, let's evaluate each one.

1. **\( 2^3 \cdot 2^2 \cdot 2^0 \)**:
- This expression can be simplified.
- \( 2^3 \cdot 2^2 \cdot 2^0 = 2^{3+2+0} = 2^5 = 32 \).
- Not equivalent to -64.

2. **\( \frac{2^8 - 2^4}{2^2} \)**:
- Calculate \( 2^8 - 2^4 \):
- \( 2^8 = 256 \) and \( 2^4 = 16 \), so \( 256 - 16 = 240 \).
- Now, divide by \( 2^2 \):
- \( \frac{240}{2^2} = \frac{240}{4} = 60 \).
- Not equivalent to -64.

3. **\( -2^{-2} \cdot (2^8) \)**:
- First, calculate \( -2^{-2} \):
- \( -2^{-2} = -\frac{1}{2^2} = -\frac{1}{4} \).
- Now multiply by \( 2^8 \):
- \( -\frac{1}{4} \cdot 256 = -64 \).
- This expression is equivalent to -64.

4. **\( -\left(\frac{1}{4}\right)^{-3} \)**:
- This can be simplified:
- \( \left(\frac{1}{4}\right)^{-3} = 4^3 = 64 \), therefore \( -\left(\frac{1}{4}\right)^{-3} = -64 \).
- This expression is also equivalent to -64.

5. **\( 4^{-3} \)**:
- Calculate \( 4^{-3} \):
- \( 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \).
- Not equivalent to -64.

The two expressions that are equivalent to -64 are:
- \( -2^{-2} \cdot (2^8) \)
- \( -\left(\frac{1}{4}\right)^{-3} \)