Question

Represent the following expression as a power of the number a where a does not equal zero.
((a to the power of -2) to the power of -1) to the power of -1 divided by (a divided by a to the power of -1) to the power of 2.

Answer 1

To simplify the expression \(\left(\left(a^{-2}\right)^{-1}\right)^{-1} \div \left(\frac{a}{a^{-1}}\right)^{2}\), we can follow these steps:

### Step 1: Simplify the numerator

We start with the part \(\left(\left(a^{-2}\right)^{-1}\right)^{-1}\):

1. \(\left(a^{-2}\right)^{-1} = a^{2}\) (using the property \(x^{-n} = \frac{1}{x^n}\)).

2. So, \(\left(a^{-2}\right)^{-1} = a^{2}\).

3. Now we raise \(a^{2}\) to the power of \(-1\):
\[
\left(a^{2}\right)^{-1} = a^{-2}.
\]

4. Finally, we raise \(a^{-2}\) to the power of \(-1\):
\[
\left(a^{-2}\right)^{-1} = a^{2}.
\]

Thus, the numerator simplifies to \(a^{2}\).

### Step 2: Simplify the denominator

Now we simplify the denominator \(\left(\frac{a}{a^{-1}}\right)^{2}\):

1. The fraction can be simplified:
\[
\frac{a}{a^{-1}} = a \cdot a^{1} = a^{1 + 1} = a^{2}.
\]

2. Now raise \(a^{2}\) to the power of \(2\):
\[
\left(a^{2}\right)^{2} = a^{4}.
\]

### Step 3: Complete the division

Now we can express the original expression:

\[
\frac{a^{2}}{a^{4}}.
\]

Using the property of exponents \(\frac{x^{m}}{x^{n}} = x^{m-n}\):

1. We can simplify the expression:
\[
a^{2 - 4} = a^{-2}.
\]

### Final Answer

Thus, the entire expression simplifies to:

\[
a^{-2}.
\]