Without expanding fully, what is $(\surd 5+\surd 3{)}^2+(\surd 5-\surd 3{)}^2$?

Expand both brackets: $(\surd 5+\surd 3{)}^2=5+2$√15 + 3 = 8 + 2√15, and $(\surd 5-\surd 3{)}^2=5-2$√15 + 3 = 8 − 2√15. Adding them, the surd parts cancel and 16 is left. The quick route is the identity $(a+b{)}^2+(a-b{)}^2=2(a^2+b^2)=2(5+3)=16$. Answering 8 evaluates only one bracket, and 16 + 4√15 comes from adding the two middle terms instead of noticing that they have opposite signs.