<u>ANSWER: </u>
Seven square root three plus two square root nine is irrational and the value is 18.12
<u>SOLUTION:
</u>
Given, seven square root three plus two square root nine
= seven square root three + two square root nine
![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B%3D7%20%5Csqrt%5B2%5D%7B3%7D%2B2%20%5Csqrt%5B2%5D%7B9%7D%7D%20%5C%5C%20%7B%3D7%20%5Csqrt%5B2%5D%7B3%7D%2B2%20%5Csqrt%5B2%5D%7B3%5E%7B2%7D%7D%7D%20%5C%5C%20%7B%3D7%20%5Csqrt%5B2%5D%7B3%7D%2B2%20%5Ctimes%203%7D%20%5C%5C%20%7B%3D7%20%5Csqrt%5B2%5D%7B3%7D%2B6%7D%5Cend%7Barray%7D)
We know that,
is irrational as it cannot be expressed as ratio of two integers.
Now,
is also irrational, because sum of rational and irrational is always irrational.
![\begin{array}{l}{\text { On solving } 7 \sqrt[2]{3}+6,} \\ {7 \sqrt[2]{3}+6=7(1.732)+6=18.12}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B%5Ctext%20%7B%20On%20solving%20%7D%207%20%5Csqrt%5B2%5D%7B3%7D%2B6%2C%7D%20%5C%5C%20%7B7%20%5Csqrt%5B2%5D%7B3%7D%2B6%3D7%281.732%29%2B6%3D18.12%7D%5Cend%7Barray%7D)
Hence, seven square root three plus two square root nine is irrational.