Select all of the expressions that are equivalent to 323−2

Question

Responses

3−4
3 to the negative 4 power

34
3

-12
-12

181
1 over 81

81
81

12

Answer 1

To determine the equivalency of the expression \( 323^{-2} \), let's first rewrite it in a simpler form.

The expression \( 323^{-2} \) can be interpreted as: \[ 323^{-2} = \frac{1}{323^2} \]

Now, let's evaluate \( 323 \):

• We can express \( 323 \) in a different way if necessary, but since we are more interested in comparing equivalent expressions, we can leave \( 323 \) as \( 3^2 \cdot 10 + 2 \), which doesn't simplify very easily. Instead, let's focus on the exponent: \[ 323^{-2} = \frac{1}{3^{2 \cdot 2}} = \frac{1}{3^{4}} = 3^{-4} \]

Now, let's check potential equivalency with the given options:

1. 3−4 (3 to the negative 4 power):

   • This is equivalent to \( 3^{-4} \), which we found to be equivalent to \( 323^{-2} \).

2. 34 (3):

   • This does not match \( 323^{-2} \).

3. -12:

   • This does not match \( 323^{-2} \).

4. 181 (1 over 81):

   • This is the same as \( \frac{1}{81} \) or \( 3^{-4} \), since \( 81 = 3^4 \). This is equivalent to \( 323^{-2} \).

5. 81:

   • This does not match \( 323^{-2} \).

6. 12:

   • This does not match \( 323^{-2} \).

Thus, the equivalent expressions to \( 323^{-2} \) from the provided list are:

1. \( 3^{-4} \)
2. \( \frac{1}{81} \) (which is represented by \( 181 \))

Final selections:

• 3−4
• 181