Solve n^{n + 1/4} / \sqrt{2} e^{n - 2\sqrt{n} + 1/2}

Evaluate

\frac{n ^{n + \frac{1}{4}}}{2} \times e^{n - 2 n + \frac{1}{2}}

To find the roots of the expression,set the expression equal to 0

\frac{n ^{n + \frac{1}{4}}}{2} \times e^{n - 2 n + \frac{1}{2}} = 0

Find the domain

\frac{n ^{n + \frac{1}{4}}}{2} \times e^{n - 2 n + \frac{1}{2}} = 0, n > 0

Calculate

\frac{n ^{n + \frac{1}{4}}}{2} \times e^{n - 2 n + \frac{1}{2}} = 0

Multiply the terms

\frac{n ^{n + \frac{1}{4}} e ^{n - 2 n + \frac{1}{2}}}{2} = 0

Simplify

n^{n + \frac{1}{4}} e^{n - 2 n + \frac{1}{2}} = 0

Separate the equation into 2 possible cases

n^{n + \frac{1}{4}} = 0 e^{n - 2 n + \frac{1}{2}} = 0

Solve the equation

More Steps

n = 0 e^{n - 2 n + \frac{1}{2}} = 0

Since the left-hand side is always positive,and the right-hand side is always 0,the statement is false for any value of n

n = 0 n \in / R

Find the union

n = 0

Check if the solution is in the defined range

n = 0, n > 0

Solution

n \in \emptyset

N^{n + 1/4} / \sqrt{2} e^{n 2\sqrt{n} + 1/2}