Which expressions are equivalent to 4^{-3} 4^{-1} ? Choose 2 answers:

To simplify the expression ( 4^{-3} 4^{-1} ), we can use the property of exponents that states ( a^m cdot a^n = a^{m+n} ).

1. Combine the exponents:

[

4^{-3} cdot 4^{-1} = 4^{-3 + (-1)} = 4^{-4}

]

Now, let’s check each option to find equivalent expressions to ( 4^{-4} ):

A. ( frac{4^{1}}{4^{3}} ):

[

frac{4^{1}}{4^{3}} = 4^{1-3} = 4^{-2} quad (text{Not equivalent})

]

B. ( frac{1}{4^{2}} ):

[

frac{1}{4^{2}} = 4^{-2} quad (text{Not equivalent})

]

C. ( 4^{3} cdot 4^{1} ):

[

4^{3} cdot 4^{1} = 4^{3+1} = 4^{4} quad (text{Not equivalent to } 4^{-4})

]

D. ( (4^{-1})^{-3} ):

[

(4^{-1})^{-3}