Question

Which expressions are equivalent to StartFraction 1 Over 81 EndFraction? Check all that apply.
8-1
9-2
StartFraction 9 Superscript 8 Over 9 Superscript 4 EndFraction
StartFraction 9 Superscript 8 Over 9 Superscript 10 EndFraction
9-6 · 93
95 · 9-7

Answer 1

To determine which expressions are equivalent to \( \frac{1}{81} \), let’s analyze each provided expression.

1. **\( 8^{-1} \)**:
\[
8^{-1} = \frac{1}{8}
\]
This is not equal to \( \frac{1}{81} \).

2. **\( 9^{-2} \)**:
\[
9^{-2} = \frac{1}{9^2} = \frac{1}{81}
\]
This expression is equivalent to \( \frac{1}{81} \).

3. **\( \frac{9^8}{9^4} \)**:
\[
\frac{9^8}{9^4} = 9^{8-4} = 9^4
\]
We know \( 9^4 = 81 \), thus \( \frac{9^8}{9^4} = 81 \), which is not equal to \( \frac{1}{81} \).

4. **\( \frac{9^8}{9^{10}} \)**:
\[
\frac{9^8}{9^{10}} = 9^{8-10} = 9^{-2} = \frac{1}{9^2} = \frac{1}{81}
\]
This expression is equivalent to \( \frac{1}{81} \).

5. **\( 9^{-6} \cdot 9^3 \)**:
\[
9^{-6} \cdot 9^3 = 9^{-6+3} = 9^{-3} = \frac{1}{9^3}
\]
Since \( 9^3 = 729 \), \( \frac{1}{9^3} \) is not equal to \( \frac{1}{81} \).

6. **\( 9^5 \cdot 9^{-7} \)**:
\[
9^5 \cdot 9^{-7} = 9^{5-7} = 9^{-2} = \frac{1}{9^2} = \frac{1}{81}
\]
This expression is also equivalent to \( \frac{1}{81} \).

In summary, the expressions that are equivalent to \( \frac{1}{81} \) are:

- \( 9^{-2} \)
- \( \frac{9^8}{9^{10}} \)
- \( 9^5 \cdot 9^{-7} \)

So the correct answers are:
- \( 9^{-2} \)
- \( \frac{9^8}{9^{10}} \)
- \( 9^5 \cdot 9^{-7} \)