Alright, let's solve the equation step-by-step:
[tex]\[4^x + \frac{1}{x^x} = 16 \times \frac{1}{16}\][/tex]
First, let's simplify the right-hand side of the equation:
[tex]\[16 \times \frac{1}{16} = 1\][/tex]
So, the equation becomes:
[tex]\[4^x + \frac{1}{x^x} = 1\][/tex]
We need to solve for [tex]\(x\)[/tex]. Let’s consider the possible values of [tex]\(x\)[/tex]:
### Case 1: [tex]\( x = 0 \)[/tex]
[tex]\(4^0 + \frac{1}{0^0}\)[/tex]
However, [tex]\(0^0\)[/tex] is not well-defined in some contexts, but in many mathematical conventions, [tex]\(0^0 = 1\)[/tex]. Therefore:
[tex]\[4^0 + \frac{1}{1} = 1 + 1 = 2\][/tex]
So, [tex]\(x = 0\)[/tex] does not satisfy the equation.
### Case 2: [tex]\( x = -1 \)[/tex]
[tex]\[4^{-1} + \frac{1}{(-1)^{-1}} = \frac{1}{4} + \frac{1}{\frac{1}{-1}} = \frac{1}{4} + (-1) = -\frac{3}{4}\][/tex]
Clearly, this does not satisfy the equation either.
### Case 3: [tex]\( x = 1 \)[/tex]
[tex]\[4^1 + \frac{1}{1^1} = 4 + 1 = 5\][/tex]
So, [tex]\(x = 1\)[/tex] does not satisfy the equation.
### Case 4: [tex]\( x = -2 \)[/tex]
[tex]\[4^{-2} + \frac{1}{(-2)^{-2}} = \frac{1}{16} + \frac{1}{\frac{1}{4}} = \frac{1}{16} + 4 = 4.0625\][/tex]
This clearly does not satisfy the equation.
### Further Analysis:
From the analysis above, the trial-and-error method does not provide a solution immediately. To simplify, we may also use numerical or graphical methods to find the exact value if simple algebraic methods become too cumbersome. However, other special values or transformations don't immediately simplify the equation any more clearly.
To solve the equation without finding an analytic solution, it may require more advanced techniques like numerical solving methods. Here let's introspect whether feasible values can be identified:
- If symmetry is considered, a transformed value [tex]\( \frac{1}{x^x} \)[/tex] may only work for reasonable intended values.
Upon further inspection and hypothesizing around values close to the tried solutions might indicate no standard algebraically simple answer.
As solving by trial, graphical plotting of [tex]\( y=4^x+\frac{1}{x^x} -1\)[/tex] with specialized mathematical approaches or software ensures finding roots within combinations of special functions tailored contexts beyond elementary scope.
Thus without stepwise numerical solving boundary, standard algebraic finding non-trivial and special verification transformed no direct-root identifiable algebraic easy proof bounds selection. Exact numerical verification gains standardization.
