Question

Which expression leaves you with the smallest number on the number line? (1 point) Responses 10/5⋅10−4 10 to the 5th power times 10 to the negative 4 power 10^8/10^10 the fraction with numerator 10 to the 8th power and denominator 10 to the 10th power (10−2)−2 open paren 10 to the negative 2 power close paren to the negative 2 power 10^8/10^8

Answer 1

To determine which expression results in the smallest number, let's evaluate each option:

1. **\( \frac{10}{5} \cdot 10^{-4} \)**
\[
\frac{10}{5} = 2 \quad \text{so} \quad 2 \cdot 10^{-4} = 0.0002
\]

2. **\( 10^5 \cdot 10^{-4} \)**
\[
10^5 \cdot 10^{-4} = 10^{5 - 4} = 10^1 = 10
\]

3. **\( \frac{10^8}{10^{10}} \)**
\[
\frac{10^8}{10^{10}} = 10^{8 - 10} = 10^{-2} = 0.01
\]

4. **\( (10^{-2})^{-2} \)**
\[
(10^{-2})^{-2} = 10^{-2 \cdot (-2)} = 10^4 = 10000
\]

5. **\( \frac{10^8}{10^8} \)**
\[
\frac{10^8}{10^8} = 10^{8 - 8} = 10^0 = 1
\]

Now let's summarize the results:
- \( 10/5 \cdot 10^{-4} = 0.0002 \)
- \( 10^5 \cdot 10^{-4} = 10 \)
- \( \frac{10^8}{10^{10}} = 0.01 \)
- \( (10^{-2})^{-2} = 10000 \)
- \( \frac{10^8}{10^8} = 1 \)

The smallest number among those calculations is \( 0.0002 \) from the expression \( \frac{10}{5} \cdot 10^{-4} \).

Therefore, the expression that leaves you with the smallest number on the number line is:
**\( \frac{10}{5} \cdot 10^{-4} \)**.