Question

Represent the following expression as a power of the number a where a does not equal zero.
(a to the power of -1 times a to the power of -2)to the power of -3

Answer 1

To simplify the expression \((a^{-1} \cdot a^{-2})^{-3}\) and represent it as a power of \(a\), we can follow these steps:

1. First, use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\):

\[
a^{-1} \cdot a^{-2} = a^{-1 + (-2)} = a^{-3}
\]

2. Now, we can substitute this back into the expression:

\[
(a^{-3})^{-3}
\]

3. Next, apply the power of a power property, which states \((a^m)^n = a^{m \cdot n}\):

\[
(a^{-3})^{-3} = a^{-3 \cdot (-3)} = a^{9}
\]

Thus, the expression \((a^{-1} \cdot a^{-2})^{-3}\) can be represented as:

\[
\boxed{a^9}
\]