Question

Which expressions are equivalent to 1/8?

2^2*2
2^4/2^-7
2^-3
2^8/2^5
2^0/2^-3
1/2^7*2^-4

Answer 1

To determine which expressions are equivalent to \( \frac{1}{8} \), we can first express \( \frac{1}{8} \) as a power of 2. Since \( 8 = 2^3 \), we have:

\[
\frac{1}{8} = \frac{1}{2^3} = 2^{-3}
\]

Now, let's evaluate each expression to see if they are equal to \( 2^{-3} \).

1. **\( 2^2 * 2 \)**:
\[
2^2 * 2 = 2^2 * 2^1 = 2^{2+1} = 2^3
\]
(Not equivalent)

2. **\( \frac{2^4}{2^{-7}} \)**:
\[
\frac{2^4}{2^{-7}} = 2^{4 - (-7)} = 2^{4 + 7} = 2^{11}
\]
(Not equivalent)

3. **\( 2^{-3} \)**:
\[
2^{-3} \text{ is equivalent to } \frac{1}{8}.
\]
(Equivalent)

4. **\( \frac{2^8}{2^5} \)**:
\[
\frac{2^8}{2^5} = 2^{8 - 5} = 2^3
\]
(Not equivalent)

5. **\( \frac{2^0}{2^{-3}} \)**:
\[
\frac{2^0}{2^{-3}} = 2^{0 - (-3)} = 2^{0 + 3} = 2^3
\]
(Not equivalent)

6. **\( \frac{1}{2^7} * 2^{-4} \)**:
\[
\frac{1}{2^7} * 2^{-4} = 2^{-7} * 2^{-4} = 2^{-7 + (-4)} = 2^{-11}
\]
(Not equivalent)

Based on our evaluations, the only expression that is equivalent to \( \frac{1}{8} \) is:

\[
2^{-3}
\]