Put the expressions in order from least to greatest.

To compare the expressions and order them from least to greatest, let’s simplify each one step by step:

1. ( frac{11^4}{11^{11}} ): Using the properties of exponents, we get ( 11^{4-11} = 11^{-7} ).

2. ( frac{1}{11^{-4}} ): This can be rewritten as ( 11^{4} ) (because ( frac{1}{a^{-b}} = a^b )).

3. ( (11^{-3})^{-3} ): Again using the properties of exponents, we find ( 11^{-3 cdot -3} = 11^{9} ).

4. ( 11^5 cdot 11^2 ): Applying the product of powers rule gives ( 11^{5+2} = 11^{7} ).

Now, we can summarize the simplified expressions:

– ( 11^{-7} )

– ( 11^{4} )

– ( 11^{9} )

– ( 11^{7} )

Next, we can order the exponents from least to greatest:

– ( 11^{-7} ) (smallest)

– ( 11^{4} )

– ( 11^{7} )

– ( 11^{9} ) (largest)

Putting it all