Sure! Let's solve the given expression step-by-step:
Expression to solve:
[tex]\[ 952 + 3 \sqrt{2} \times \frac{252 + 2 \sqrt{2}}{12 + 2 \sqrt{3}} \][/tex]
Step 1: Simplify the numerator and denominator of the fraction.
The numerator is:
[tex]\[ 252 + 2 \sqrt{2} \][/tex]
Evaluating this:
[tex]\[ 252 + 2 \sqrt{2} \approx 254.82842712474618 \][/tex]
The denominator is:
[tex]\[ 12 + 2 \sqrt{3} \][/tex]
Evaluating this:
[tex]\[ 12 + 2 \sqrt{3} \approx 15.464101615137753 \][/tex]
Step 2: Compute the fraction value.
[tex]\[
\frac{252 + 2 \sqrt{2}}{12 + 2 \sqrt{3}} \approx \frac{254.82842712474618}{15.464101615137753} \approx 16.47870878417506
\][/tex]
Step 3: Multiply this value by [tex]\(3 \sqrt{2}\)[/tex].
First, evaluate [tex]\(3 \sqrt{2}\)[/tex]:
[tex]\[ 3 \sqrt{2} \approx 3 \times 1.4142135623730951 \approx 4.242640687119285 \][/tex]
Now multiply:
[tex]\[
3 \sqrt{2} \times \frac{252 + 2 \sqrt{2}}{12 + 2 \sqrt{3}} \approx 4.242640687119285 \times 16.47870878417506 \approx 69.91324035893108
\][/tex]
Step 4: Add this result to 952.
[tex]\[
952 + 69.91324035893108 \approx 1021.913240358931
\][/tex]
So, the final result of the given expression is:
[tex]\[ \boxed{1021.913240358931} \][/tex]
