Question

give an example each of two irrational numbers whose (i)difference is a rational(ii)sum is an irrational(iii)product is a rational(iv)quotient is an irrational number

Answer 1

Dear student,

$$\begin{gathered} i)Differenceisarationalnumber \\ 4+\sqrt{3}and2+\sqrt{3}\left[4+\sqrt{3}-2-\sqrt{3}=2whichisarationalnumber\right] \\ ii)Sumisanirrationalnumber \\ \sqrt{2}and\sqrt{5}\left[\left(\sqrt{2}\right)+\left(\sqrt{5}\right)=\sqrt{2}+\sqrt{5}whichisirrationalnumber\right] \\ iii)Productisarationalnumber \\ 6+\sqrt{7}and6-\sqrt{7}\left[\left(6+\sqrt{7}\right)\times \left(6-\sqrt{7}\right)=6^2-{\left(\sqrt{7}\right)}^2=36-7=29whichisarationalnumber\right] \\ iv)quotientisanirrationalnumber \\ 4\sqrt{6}and2\sqrt{3}\left[\frac{4\sqrt{6}}{2\sqrt{3}}=\frac{4\sqrt{2}\sqrt{3}}{2\sqrt{3}}=2\sqrt{2}whichisirrationalnumber\right] \end{gathered}$$
Regards