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

Question

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

4−3
4 to the negative 3 power

−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

Answer 1

To determine which expressions are equivalent to -64, let's evaluate each of the options step by step:

1. \( 2^3 \cdot 2^2 \cdot 2^0 \):

   • This simplifies to \( 2^{3+2+0} = 2^5 = 32 \). This is not equivalent to -64.

2. \( \frac{2^8 - 2^4}{2^2} \):

   • Calculate \( 2^8 = 256 \) and \( 2^4 = 16 \):
   • Then, \( 256 - 16 = 240 \).
   • Now divide by \( 2^2 = 4 \):
   • \( \frac{240}{4} = 60 \). This is not equivalent to -64.

3. \( 4^{-3} \):

   • This is equivalent to \( \frac{1}{4^3} = \frac{1}{64} \). This is not equivalent to -64.

4. \( -2^{-2} \cdot (2^8) \):

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

5. \( -\left(\frac{1}{4}\right)^{-3} \):

   • The expression \( \left(\frac{1}{4}\right)^{-3} = 4^3 = 64 \).
   • So, \( -64 \). This is equivalent to -64.

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

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