Let's solve the given expression step-by-step:
Given the expression:
[tex]\[ x^{\cosh^2\left(\frac{x}{a}\right)} \][/tex]
1. Identify the components:
- [tex]\( x \)[/tex]: The base of the expression
- [tex]\( \cosh\left(\frac{x}{a}\right) \)[/tex]: The hyperbolic cosine function evaluated at [tex]\(\frac{x}{a}\)[/tex]
- [tex]\( \cosh^2\left(\frac{x}{a}\right) \)[/tex]: Squaring the result of the hyperbolic cosine function
2. Understanding the Hyperbolic Cosine Function:
- The hyperbolic cosine function is defined as:
[tex]\[ \cosh(y) = \frac{e^y + e^{-y}}{2} \][/tex]
- In this problem, the variable [tex]\( y \)[/tex] is replaced with [tex]\(\frac{x}{a}\)[/tex], so we have:
[tex]\[ \cosh\left(\frac{x}{a}\right) = \frac{e^{\frac{x}{a}} + e^{-\frac{x}{a}}}{2} \][/tex]
3. Square of Hyperbolic Cosine Function:
- Squaring the hyperbolic cosine function:
[tex]\[ \cosh^2\left(\frac{x}{a}\right) = \left(\frac{e^{\frac{x}{a}} + e^{-\frac{x}{a}}}{2}\right)^2 \][/tex]
Simplifying the square:
[tex]\[ \cosh^2\left(\frac{x}{a}\right) = \left(\frac{e^{\frac{x}{a}} + e^{-\frac{x}{a}}}{2}\right) \times \left(\frac{e^{\frac{x}{a}} + e^{-\frac{x}{a}}}{2}\right) \][/tex]
[tex]\[ \cosh^2\left(\frac{x}{a}\right) = \frac{(e^{\frac{x}{a}} + e^{-\frac{x}{a}})^2}{4} \][/tex]
Expanding the numerator:
[tex]\[ \cosh^2\left(\frac{x}{a}\right) = \frac{e^{\frac{2x}{a}} + 2 + e^{-\frac{2x}{a}}}{4} \][/tex]
4. Final Expression:
- Substitute the squared hyperbolic cosine back into the original expression:
[tex]\[ x^{\cosh^2\left(\frac{x}{a}\right)} = x^{\frac{e^{\frac{2x}{a}} + 2 + e^{-\frac{2x}{a}}}{4}} \][/tex]
However, recognizing the structure and arriving directly to the answer, we arrive at the simplified form:
[tex]\[ x^{\left(\cosh\left(\frac{x}{a}\right)\right)^2} \][/tex]
Thus, the final expression we were asked to find is:
[tex]\[ x^{\cosh^2 \left(\frac{x}{a}\right)} \][/tex]
This process breaks down the components of the given expression step-by-step to understand the underlying functions and operations.
