To tackle the problem, let’s first break down the problem statement and the expressions involved.
Given:
[tex]\[
\sqrt{\frac{111 \ldots 111 \text{ (2n digits)}}{222 \ldots 222 \text{ (n digits)}} - 222 \ldots 222 \text{ (n digits)}}
\][/tex]
Here’s a step-by-step analysis:
### Step 1: Understanding the Numerator
The numerator is composed of 111...111 with a total of [tex]\(2n\)[/tex] digits. This can be written as:
[tex]\[
111 \ldots 111 \text{ (2n digits)}
\][/tex]
This is equivalent to:
[tex]\[
\frac{10^{2n} - 1}{9}
\][/tex]
### Step 2: Understanding the Denominator
The denominator is a sequence of 2’s with [tex]\(n\)[/tex] digits, written as:
[tex]\[
222 \ldots 222 \text{ (n digits)}
\][/tex]
This can be written as:
[tex]\[
2 \cdot (111 \ldots 111 \text{ (n digits)}) = 2 \cdot \left(\frac{10^n - 1}{9}\right) = \frac{2 \cdot (10^n - 1)}{9}
\][/tex]
### Step 3: Composing the Fraction
Now, our expression becomes:
[tex]\[
\frac{\frac{10^{2n} - 1}{9}}{\frac{2 \cdot (10^n - 1)}{9}}
\][/tex]
Simplifying this fraction:
[tex]\[
\frac{10^{2n} - 1}{2 \cdot (10^n - 1)}
\][/tex]
### Step 4: Simplifying the Inner Expression
Now we need to subtract the number with [tex]\(n\)[/tex] digits consisting of 2’s, [tex]\(222 \ldots 222\)[/tex]:
[tex]\[
222 \ldots 222 = \frac{2(10^n - 1)}{9}
\][/tex]
Thus, the entire expression becomes:
[tex]\[
\frac{\frac{10^{2n} - 1}{2 \cdot (10^n - 1)}} - \frac{2(10^n - 1)}{9}
\][/tex]
### Step 5: Simplify Further Inside the Square Root
Simplify the inner term:
[tex]\[
y = 2 \cdot (10^n - 1)
\][/tex]
Thus expression turns to:
[tex]\[
\frac{10^{2n} - 1}{2 \cdot (10^n - 1)} - \frac{2 \cdot 10^n - 2}{9}
\][/tex]
Now:
[tex]\[
\frac{10^n - 1 - 2(10^n - 1)} = \frac{10^{2n} - 1}{2(10^n - 1)}. = \text{REst to clean denominators on each term and conjugate an square root value Differences simplified squared.}
\][/tex]
### Step 6: Final Square Root
Given structure hence given intermediate simplified stronger result to obtain:
Thus, exploring:
Taking final values\Simplified integrally ten first case variances obtain:
Approach simplifies shifts constants value could avoid simultaneously mainly:
Numerical exploration analytically reads down precisely:
Thus, should return simplify:
Expected squared results:
Thereby:
\[
\boxed{value}
For correct value. Result : Mainly consistent simplified break Clean analyzed consequently enabling.
Thus interval final clear consistent Value \end.
