Which of these statements are true for all a, b > 0? Explain your conjecture. (i). 2(a + b) = 2a + 2b (ii) a+b/2= a/2+b/2 (iii) √a+b= √a+√b

Answer:Statement (i) , (ii) are TRUE  and The statement (iii) is FALSE.Step-by-step explanation:Given :  a , b are strictly greater than  0. Now, let us take each statements.(i) 2(a +b) =  2a + 2b Yes, the given statement is TRUE, as by DISTRIBUTIVE PROPERTY we get that x (y + z) = xy + xz(ii)[tex]\frac{a + b}{2}  = \frac{a}{2}  +   \frac{b}{2}[/tex]Yes, the given statement is TRUE, as by DISTRIBUTIVE PROPERTY we get that [tex]\frac{m + n}{k}  = \frac{m}{k}  +   \frac{n}{k}[/tex](iii[tex]\sqrt{(a +b)}  = \sqrt{a}  + \sqrt{b}[/tex]Here, the given statement is FALSE. Because, if we have a = 2 and b = 3, then[tex]\sqrt{(2+ 3} ) = \sqrt{5}  =  2.23\\\sqrt{2}  + \sqrt{3}  = 1.41 +  1.73 =  3.14\\[/tex]and 2.23 ≠ 3.24So,[tex]\sqrt{(a +b)}  \neq \sqrt{a}  + \sqrt{b}[/tex]