Let's solve the given equation step-by-step:
[tex]\[ \sqrt{\frac{7^x + 7^{14}}{7^2 + 7^x}} = 7 \][/tex]
### Step 1: Remove the square root by squaring both sides
We start by squaring both sides of the equation to eliminate the square root:
[tex]\[
\left( \sqrt{\frac{7^x + 7^{14}}{7^2 + 7^x}} \right)^2 = 7^2
\][/tex]
This simplifies to:
[tex]\[
\frac{7^x + 7^{14}}{7^2 + 7^x} = 49
\][/tex]
### Step 2: Simplify the equation
Now, we multiply both sides of the equation by the denominator to clear the fraction:
[tex]\[
7^x + 7^{14} = 49 \cdot (7^2 + 7^x)
\][/tex]
### Step 3: Distribute the 49 on the right side
Distribute 49 to both terms inside the parentheses on the right-hand side:
[tex]\[
7^x + 7^{14} = 49 \cdot 7^2 + 49 \cdot 7^x
\][/tex]
[tex]\[
7^x + 7^{14} = 49 \cdot 49 + 49 \cdot 7^x
\][/tex]
[tex]\[
7^x + 7^{14} = 2401 + 49 \cdot 7^x
\][/tex]
### Step 4: Collect like terms
Move all terms involving [tex]\(7^x\)[/tex] to one side of the equation:
[tex]\[
7^x + 7^{14} - 49 \cdot 7^x = 2401
\][/tex]
Combine the [tex]\(7^x\)[/tex] terms on the left side:
[tex]\[
7^x - 49 \cdot 7^x + 7^{14} = 2401
\][/tex]
Factor out [tex]\(7^x\)[/tex] from the left side:
[tex]\[
7^x (1 - 49) + 7^{14} = 2401
\][/tex]
This simplifies to:
[tex]\[
-48 \cdot 7^x + 7^{14} = 2401
\][/tex]
### Step 5: Move constant term to the right side
Move [tex]\(7^{14}\)[/tex] to the right side:
[tex]\[
-48 \cdot 7^x = 2401 - 7^{14}
\][/tex]
### Step 6: Simplify and solve for [tex]\(x\)[/tex]
Notice that [tex]\(7^{14} > 2401\)[/tex]. To solve for [tex]\(x\)[/tex], we can look at the previously obtained result.
Given that the actual solution simplifies to:
[tex]\[
x = \frac{\log(14129647301)}{\log(7)}
\][/tex]
This represents the exact value where the original equation balances out when the terms are logarithmically expressed.
So, the solution to the equation
[tex]\[ \sqrt{\frac{7^x + 7^{14}}{7^2 + 7^x}} = 7 \][/tex]
is
[tex]\[ x = \frac{\log(14129647301)}{\log(7)} \][/tex]
