Question

Tell whether each expression in the table is equivalent to 5^2, 5^-2, or neither.
A.) (5^-1)^2
B.) (5^-1)^-1
C.) (1/5)^2
D.) (1/5)^-2

Answer 1

To determine whether each expression is equivalent to \(5^2\), \(5^{-2}\), or neither, we will simplify each expression:

A.) \((5^{-1})^2\)
\[
(5^{-1})^2 = 5^{-2}
\]
This is equivalent to \(5^{-2}\).

B.) \((5^{-1})^{-1}\)
\[
(5^{-1})^{-1} = 5^{(-1)(-1)} = 5^1
\]
This is neither \(5^2\) nor \(5^{-2}\).

C.) \((1/5)^2\)
\[
(1/5)^2 = \frac{1^2}{5^2} = \frac{1}{5^2} = 5^{-2}
\]
This is equivalent to \(5^{-2}\).

D.) \((1/5)^{-2}\)
\[
(1/5)^{-2} = (5^{-1})^{-2} = 5^{(-1)(-2)} = 5^2
\]
This is equivalent to \(5^2\).

Now let's summarize:

- A.) \(5^{-2}\)
- B.) Neither
- C.) \(5^{-2}\)
- D.) \(5^2\)