The value of `sqrte` up to four decimals is

Options

A. 1.5237
B. 1.2589
C. 1.6484
D. `1.9190`

Answer

To find the value of \(\sqrt{e}\) up to four decimal places, we can use the exponential series expansion for \(e^x\).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to calculate \(\sqrt{e}\). We know that \(\sqrt{e} = e^{1/2}\).

2. **Using the Exponential Series**:
The exponential function can be expressed as a series:
\[ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \ldots \]
For our case, we will substitute \(x = \frac{1}{2}\):
\[ e^{1/2} = 1 + \frac{1/2}{1!} + \frac{(1/2)^2}{2!} + \frac{(1/2)^3}{3!} + \frac{(1/2)^4}{4!} + \ldots \]

3. **Calculating Each Term**:
- The first term is \(1\).
- The second term is \(\frac{1/2}{1!} = \frac{1}{2} = 0.5\).
- The third term is \(\frac{(1/2)^2}{2!} = \frac{1/4}{2} = \frac{1}{8} = 0.125\).
- The fourth term is \(\frac{(1/2)^3}{3!} = \frac{1/8}{6} = \frac{1}{48} \approx 0.02083\).
- The fifth term is \(\frac{(1/2)^4}{4!} = \frac{1/16}{24} = \frac{1}{384} \approx 0.00260\).

4. **Summing the Terms**:
Now we will sum these terms:
\[ 1 + 0.5 + 0.125 + 0.02083 + 0.00260 \approx 1.64843 \]

5. **Rounding to Four Decimal Places**:
Rounding \(1.64843\) to four decimal places gives us \(1.6484\).

6. **Conclusion**:
Therefore, the value of \(\sqrt{e}\) up to four decimal places is:
\[ \sqrt{e} \approx 1.6484 \]

### Final Answer:
The value of \(\sqrt{e}\) up to four decimals is **1.6484**.