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Select all the expressions that are equivalent to 8 raised to the power of negative 5 divided by 2 raised to the power of negative 5.
(4 raised to the power of negative 1) raised to the power of 5
4
1 divided by 4 raised to the power of negative 5
4 raised to the power of negative 5
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. **