Select all the expressions that are equivalent to

To solve the expression ( frac{8^{-5}}{2^{-5}} ), we first simplify it:

1. Rewrite 8 as ( 2^3 ):

[

8^{-5} = (2^3)^{-5} = 2^{-15}

]

2. Now substitute back into the expression:

[

frac{8^{-5}}{2^{-5}} = frac{2^{-15}}{2^{-5}} = 2^{-15 – (-5)} = 2^{-15 + 5} = 2^{-10}

]

Now we can analyze the given options:

1. (4 raised to the power of negative 1) raised to the power of 5:

[

(4^{-1})^5 = 4^{-5} = (2^2)^{-5} = 2^{-10} quad text{(equivalent)}

]

2. 4:

( 4 = 2^{2} quad text{(not equivalent)} )

3. 1 divided by 4 raised to the power of negative 5:

[

frac{1}{4^{-5}} = 4^{5} = (2^2)^{5} = 2^{10} quad text{(not equivalent)}

]

4. **