Which two expressions are equivalent to each other? 9^-5 9^9/9^4 (9^2)^-3 9^4 · 9^-9

To find the equivalent expressions, we can simplify each one using the rules of exponents.

1. 9^-5: This is already simplified.

2. 9^9 / 9^4: Using the rule ( a^m / a^n = a^{m-n} ):

[

9^{9-4} = 9^5

]

3. (9^2)^-3: Using the power of a power rule ( (a^m)^n = a^{m cdot n} ):

[

9^{2 cdot -3} = 9^{-6}

]

4. 9^4 · 9^-9: Using the rule ( a^m cdot a^n = a^{m+n} ):

[

9^{4 + (-9)} = 9^{-5}

]

Now, to summarize:

– 9^-5 is equivalent to 9^{-5}.

– 9^9 / 9^4 simplifies to 9^5.

– (9^2)^-3 simplifies to 9^{-6}.

– 9^4 · 9^-9 simplifies to 9^{-5}.

The equivalent expressions are 9^-5 and 9^4 · 9^-9.