(i) Consider the two irrational number 2+√3 and √3−2.
Their sum =2+√3+√3−2=2√3 is an irrational number.
(ii) Consider the two irrational numbers √2 and −√2.
Their sum =√2+(−√2)=0 is a rational number.
(iii) Consider the two irrational numbers √3 and √2.
Their difference=√3−√2 is an irrational number.
(iv) Consider the two irrational numbers 5+√3 and √3−5.
Their difference =(5+√3)−(√3−5)=10 is a rational number.
(v) Consider the two irrational numbers √3 and √5.
Their product =√3×√5=√15 is an irrational number.
(vi) Consider the two irrational numbers √18 and √2.
Their product =√18×√2=√36=6 is a rational number.
(vii) Consider the two irrational numbers √15 and √3.
Their quotient =√15√3=√153=√5 is an irrational number.
(viii) Consider the two irrational numbers √75 and √3.
Their quotient =√75√3=√753=5 is a rational number.