Question

use your calculator to evaluate $e^{-2}$. round your answer to three decimal places.

a. 0.084
b. -5.436
c. -7.389
d. 0.135

Explanation:

Step1: Recall the value of \( e \)

The constant \( e \) is approximately \( 2.71828 \).

Step2: Calculate \( e^{-2} \)

Using the formula \( a^{-n}=\frac{1}{a^{n}} \), we have \( e^{-2}=\frac{1}{e^{2}} \). First, calculate \( e^{2} \approx 2.71828^{2}\approx 7.38906 \). Then, \( \frac{1}{7.38906}\approx 0.135 \)? Wait, no, wait. Wait, \( e^{-2} \) is \( \frac{1}{e^{2}} \), and \( e^{2}\approx 7.38906 \), so \( \frac{1}{7.38906}\approx 0.135 \)? Wait, no, wait, maybe I made a mistake. Wait, let's check again. Wait, \( e\approx 2.71828 \), so \( e^{2}=2.71828\times2.71828\approx 7.38906 \), then \( e^{-2}=\frac{1}{e^{2}}\approx\frac{1}{7.38906}\approx 0.135 \)? Wait, but let's use a calculator. If we use a calculator to compute \( e^{-2} \), we get approximately \( 0.1353352832 \). Rounding to three decimal places, that's \( 0.135 \)? Wait, no, wait, the options: D is 0.135, A is 0.084, B is -5.436, C is -7.389. Wait, but \( e^{-2} \) is positive, so B and C are negative, so we can eliminate them. Then between A (0.084) and D (0.135). Wait, maybe I miscalculated. Wait, let's use a calculator. Let's compute \( e^{-2} \):

Using a calculator, \( e^{-2} = \frac{1}{e^{2}} \approx \frac{1}{7.389056099} \approx 0.1353352832 \). Rounding to three decimal places, that's \( 0.135 \), which is option D? Wait, but wait, maybe the problem is \( e^{2} \)? No, the problem is \( e^{-2} \). Wait, let's check the options again. Option D is 0.135, which matches the rounded value of \( e^{-2} \approx 0.135 \). Wait, but let's confirm. So step by step:

1. Recall that \( e \approx 2.71828 \).
2. Compute \( e^{2} = e \times e \approx 2.71828 \times 2.71828 \approx 7.38906 \).
3. Compute \( e^{-2} = \frac{1}{e^{2}} \approx \frac{1}{7.38906} \approx 0.135335 \).
4. Round to three decimal places: look at the fourth decimal place, which is 3, so we round down? Wait, no, 0.135335, the third decimal is 5, the fourth is 3, so we keep the third decimal as 5? Wait, 0.135335 rounded to three decimal places: the first decimal is 1, second is 3, third is 5, fourth is 3. Since the fourth is less than 5, we don't round up the third. So 0.135. So that's option D.

Answer:

D. 0.135