To determine which given choice is equivalent to the expression [tex]\(\left(\frac{6}{22}+\frac{5}{77}\right)^2\)[/tex], let’s follow a step-by-step approach.
### Step 1: Simplify the Expression
First, we need to handle the terms inside the parentheses:
#### Simplifying [tex]\(\frac{6}{22}\)[/tex]
The fraction [tex]\(\frac{6}{22}\)[/tex] can be simplified by dividing the numerator and denominator by their greatest common divisor (GCD), which is 2:
[tex]\[
\frac{6}{22} = \frac{6 \div 2}{22 \div 2} = \frac{3}{11}
\][/tex]
#### Simplifying [tex]\(\frac{5}{77}\)[/tex]
The fraction [tex]\(\frac{5}{77}\)[/tex] is already in its simplest form because 5 and 77 have no common divisors other than 1:
[tex]\[
\frac{5}{77} = \frac{5}{77}
\][/tex]
### Step 2: Perform the Addition Inside the Parentheses
Next, let's add the simplified fractions [tex]\(\frac{3}{11}\)[/tex] and [tex]\(\frac{5}{77}\)[/tex].
To do this, we need a common denominator for the fractions. The least common multiple (LCM) of 11 and 77 is 77. Thus, we convert [tex]\(\frac{3}{11}\)[/tex] to have a denominator of 77:
[tex]\[
\frac{3}{11} = \frac{3 \times 7}{11 \times 7} = \frac{21}{77}
\][/tex]
Now, add [tex]\(\frac{21}{77}\)[/tex] and [tex]\(\frac{5}{77}\)[/tex]:
[tex]\[
\frac{21}{77} + \frac{5}{77} = \frac{21 + 5}{77} = \frac{26}{77}
\][/tex]
### Step 3: Square the Result
Now, we need to square the sum [tex]\(\frac{26}{77}\)[/tex]:
[tex]\[
\left(\frac{26}{77}\right)^2 = \frac{26^2}{77^2} = \frac{676}{5929}
\][/tex]
### Step 4: Compare to the Choices
We compare [tex]\(\frac{676}{5929}\)[/tex] with the given choices to see which one is equivalent.
1. [tex]\(\frac{1}{11^2}\left(3 + \frac{5}{7}\right)^2\)[/tex]
- Simplify inside parentheses:
[tex]\[
3 + \frac{5}{7} = \frac{21}{7} + \frac{5}{7} = \frac{26}{7}
\][/tex]
- Square it:
[tex]\[
\left(\frac{26}{7}\right)^2 = \frac{676}{49}
\][/tex]
- Multiply by [tex]\(\frac{1}{11^2}\)[/tex]:
[tex]\[
\frac{1}{11^2} \cdot \frac{676}{49} = \frac{1}{121} \cdot \frac{676}{49}
\][/tex]
[tex]\[
\frac{676}{49 \times 121} = \frac{676}{5929}
\][/tex]
This matches our expression!
2. [tex]\(\left(\frac{6}{22}\right)^2 + \left(\frac{5}{77}\right)^2\)[/tex]
[tex]\[
\left(\frac{6}{22}\right)^2 = \left(\frac{3}{11}\right)^2 = \frac{9}{121}
\][/tex]
[tex]\[
\left(\frac{5}{77}\right)^2 = \frac{25}{5929}
\][/tex]
[tex]\[
\frac{9}{121} + \frac{25}{5929}
\][/tex]
This does not match [tex]\(\frac{676}{5929}\)[/tex].
3. [tex]\(\frac{1}{9^2} = \frac{1}{81}\)[/tex]
This does not match [tex]\(\frac{676}{5929}\)[/tex].
4. [tex]\(\left(\frac{1}{2} + \frac{1}{7}\right)^2\)[/tex]
[tex]\[
\frac{1}{2} + \frac{1}{7} = \frac{7}{14} + \frac{2}{14} = \frac{9}{14}
\][/tex]
[tex]\[
\left(\frac{9}{14}\right)^2 = \frac{81}{196}
\][/tex]
This does not match [tex]\(\frac{676}{5929}\)[/tex].
Thus, the correct choice is:
[tex]\[
\boxed{\frac{1}{11^2}\left(3 + \frac{5}{7}\right)^2}
\][/tex]
