To solve the expression [tex]\(\sqrt{2}+\frac{1}{2}-\sqrt{5}+e\)[/tex], we will break it down into parts and evaluate each term individually.
1. Evaluating [tex]\(\sqrt{2}\)[/tex]:
The square root of 2 ([tex]\(\sqrt{2}\)[/tex]) is approximately 1.4142135623730951.
2. Evaluating [tex]\(\frac{1}{2}\)[/tex]:
The fraction [tex]\(\frac{1}{2}\)[/tex] is simply 0.5.
3. Evaluating [tex]\(\sqrt{5}\)[/tex]:
The square root of 5 ([tex]\(\sqrt{5}\)[/tex]) is approximately 2.23606797749979.
4. Evaluating [tex]\(e\)[/tex]:
The constant [tex]\(e\)[/tex] (Euler's number) is approximately 2.718281828459045.
Now that we have the values for each term, let's substitute them into the expression:
[tex]\[
\sqrt{2}+\frac{1}{2}-\sqrt{5}+e
\][/tex]
Substituting the approximate values, we get:
[tex]\[
1.4142135623730951 + 0.5 - 2.23606797749979 + 2.718281828459045
\][/tex]
Next, perform the addition and subtraction step-by-step:
1. First, add [tex]\(\sqrt{2}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
1.4142135623730951 + 0.5 = 1.9142135623730951
\][/tex]
2. Then, subtract [tex]\(\sqrt{5}\)[/tex] from the result:
[tex]\[
1.9142135623730951 - 2.23606797749979 = -0.3218544151266949
\][/tex]
3. Lastly, add [tex]\(e\)[/tex] to the result:
[tex]\[
-0.3218544151266949 + 2.718281828459045 = 2.3964274133323507
\][/tex]
Thus, the final value of the expression [tex]\(\sqrt{2}+\frac{1}{2}-\sqrt{5}+e\)[/tex] is approximately [tex]\(2.3964274133323507\)[/tex].
