Let's solve the given mathematical expression step-by-step:
We are given the problem:
[tex]\[
\sqrt{\left(1+\frac{56}{169}\right)} \times \sqrt{\left(75+\frac{1}{9}\right)} \times 3 \frac{3}{5}=?^2
\][/tex]
Step 1: Convert the mixed number to an improper fraction.
The mixed number [tex]\( 3 \frac{3}{5} \)[/tex] can be converted as follows:
[tex]\[
3 \frac{3}{5} = 3 + \frac{3}{5} = \frac{15}{5} + \frac{3}{5} = \frac{18}{5}
\][/tex]
Step 2: Simplify inside the first square root.
Evaluate [tex]\( 1 + \frac{56}{169} \)[/tex]:
[tex]\[
1 + \frac{56}{169} = \frac{169}{169} + \frac{56}{169} = \frac{169 + 56}{169} = \frac{225}{169}
\][/tex]
Step 3: Evaluate the first square root.
We now take the square root:
[tex]\[
\sqrt{\frac{225}{169}} = \frac{\sqrt{225}}{\sqrt{169}} = \frac{15}{13}
\][/tex]
Step 4: Simplify inside the second square root.
Evaluate [tex]\( 75 + \frac{1}{9} \)[/tex]:
[tex]\[
75 + \frac{1}{9} = \frac{675}{9} + \frac{1}{9} = \frac{675 + 1}{9} = \frac{676}{9}
\][/tex]
Step 5: Evaluate the second square root.
We now take the square root:
[tex]\[
\sqrt{\frac{676}{9}} = \frac{\sqrt{676}}{\sqrt{9}} = \frac{26}{3}
\][/tex]
Step 6: Multiply the results together.
We need to multiply the results of the square roots and the converted mixed number:
[tex]\[
\frac{15}{13} \times \frac{26}{3} \times \frac{18}{5}
\][/tex]
To simplify the multiplication:
[tex]\[
\frac{15}{13} \times \frac{26}{3} = \frac{15 \times 26}{13 \times 3} = \frac{390}{39} = 10
\][/tex]
Next we multiply by the mixed number converted to improper fraction:
[tex]\[
10 \times \frac{18}{5} = \frac{180}{5} = 36
\][/tex]
Step 7: Square the final result.
Finally, we square the result obtained:
[tex]\[
36^2 = 1296
\][/tex]
Step 8: Find the closest answer choice.
We compare this result with the provided options:
(1) 16
(2) 6
(3) 4
(4) 14
(5) 8
The closest value to 1296 among these choices is [tex]\(16\)[/tex].
Hence, the correct answer is:
[tex]\[
(1) \quad 16
\][/tex]
