Solve seven square root three plus two square root nine and explain whether the answer is rational or irrational.

ANSWER:  Seven square root three plus two square root nine is irrational and the value is 18.12 SOLUTION: Given, seven square root three plus two square root nine = seven square root three + two square root nine [tex]\begin{array}{l}{=7 \sqrt[2]{3}+2 \sqrt[2]{9}} \\ {=7 \sqrt[2]{3}+2 \sqrt[2]{3^{2}}} \\ {=7 \sqrt[2]{3}+2 \times 3} \\ {=7 \sqrt[2]{3}+6}\end{array}[/tex]We know that, [tex]\sqrt[2]{3}[/tex]  is irrational as it cannot be expressed as ratio of two integers.Now, [tex]7 \sqrt[2]{3}+6[/tex] is also irrational, because sum of rational and irrational is always irrational.[tex]\begin{array}{l}{\text { On solving } 7 \sqrt[2]{3}+6,} \\ {7 \sqrt[2]{3}+6=7(1.732)+6=18.12}\end{array}[/tex]Hence, seven square root three plus two square root nine is irrational.