To solve the equation [tex]\(4^x + \frac{1}{4^x} = 16 + \frac{1}{16}\)[/tex], let's proceed step-by-step.
### Step 1: Simplify the given equation
We start with the given equation:
[tex]\[ 4^x + \frac{1}{4^x} = 16 + \frac{1}{16} \][/tex]
### Step 2: Simplify the right-hand side
We know:
[tex]\[ 16 = 4^2 \][/tex]
[tex]\[ \frac{1}{16} = 4^{-2} \][/tex]
So the right-hand side of the equation can be written as:
[tex]\[ 16 + \frac{1}{16} = 4^2 + 4^{-2} \][/tex]
Thus, our equation simplifies to:
[tex]\[ 4^x + \frac{1}{4^x} = 4^2 + 4^{-2} \][/tex]
### Step 3: Let [tex]\( y = 4^x \)[/tex]
This substitution helps to transform the equation. Since [tex]\( y = 4^x \)[/tex], we have:
[tex]\[ \frac{1}{4^x} = \frac{1}{y} \][/tex]
Then the equation becomes:
[tex]\[ y + \frac{1}{y} = 4^2 + 4^{-2} \][/tex]
### Step 4: Solve the algebraic equation
This simplifies to:
[tex]\[ y + \frac{1}{y} = 16 + \frac{1}{16} \][/tex]
This can be written as:
[tex]\[ y + \frac{1}{y} = 16.0625 \][/tex]
To solve for [tex]\( y \)[/tex], let us consider the solutions derived from solving this equation for [tex]\( x \)[/tex]:
[tex]\[ 4^x = y \][/tex]
[tex]\[ \frac{1}{4^x} = \frac{1}{y} \][/tex]
Given the solutions to the original problem are:
[tex]\[ x = -2, 2, -2 + 4.53236014182719i, 2 + 4.53236014182719i \][/tex]
### Step 5: Verify the values
The solutions [tex]\(x = -2\)[/tex] and [tex]\(x = 2\)[/tex] can be verified in the balanced form:
[tex]\[ 4^{-2} = \frac{1}{16} \][/tex]
[tex]\[ 4^{2} = 16 \][/tex]
Substituting these back into the original equation satisfies it:
[tex]\[ 16 + \frac{1}{16} = 16.0625 \][/tex]
The complex components [tex]\( -2 + 4.53236014182719i \)[/tex] and [tex]\( 2 + 4.53236014182719i \)[/tex] suggest additional forms of the solutions which arise due to the periodic nature of the exponential function.
Thus, the complete set of solutions are:
[tex]\[ x = -2, 2, -2 + 4.53236014182719i, 2 + 4.53236014182719i \][/tex]
These represent all the solutions for the given equation:
[tex]\[ 4^x + \frac{1}{4^x} = 16 + \frac{1}{16} \][/tex]
