Alright, let's solve the given mathematical expression step-by-step.
We are given the expression:
[tex]\[
\frac{2^{x+1} + 2^x}{2^{x+1} - 2^x}
\][/tex]
First, let’s simplify the terms in the numerator and the denominator.
### Step 1: Simplify [tex]\( 2^{x+1} \)[/tex]
Recall that [tex]\( 2^{x+1} \)[/tex] can be rewritten as:
[tex]\[
2^{x+1} = 2 \cdot 2^x
\][/tex]
### Step 2: Substitute [tex]\( 2^{x+1} \)[/tex] back into the expression
Now replace [tex]\( 2^{x+1} \)[/tex] in the original expression:
[tex]\[
\frac{2 \cdot 2^x + 2^x}{2 \cdot 2^x - 2^x}
\][/tex]
### Step 3: Factor out [tex]\( 2^x \)[/tex] in both the numerator and the denominator
Factor [tex]\( 2^x \)[/tex] from both terms in the numerator and the denominator:
[tex]\[
\frac{2^x (2 + 1)}{2^x (2 - 1)}
\][/tex]
### Step 4: Simplify the fractions
Since [tex]\( 2^x \)[/tex] is common in both the numerator and the denominator, we can cancel it out:
[tex]\[
\frac{2^x (3)}{2^x (1)} = \frac{3 \cdot 2^x}{1 \cdot 2^x} = \frac{3}{1}
\][/tex]
### Step 5: Final answer
After canceling [tex]\( 2^x \)[/tex] from both the numerator and the denominator, we are left with:
[tex]\[
3
\][/tex]
So, the simplified expression is [tex]\( 3 \)[/tex].
