Question

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 1

Statement (i) , (ii) are TRUE and

The statement (iii) is FALSE.

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)
`(a + b)/(2)  = (a)/(2)  +   (b)/(2)`

Yes, the given statement is TRUE, as by DISTRIBUTIVE PROPERTY we get that
`(m + n)/(k)  = (m)/(k)  +   (n)/(k)`

(iii
`√((a +b))  = √(a)  + √(b)`

Here, the given statement is FALSE.

Because, if we have a = 2 and b = 3, then

`√((2+ 3) ) = √(5)  =  2.23\\√(2)  + √(3)  = 1.41 +  1.73 =  3.14\\`

and 2.23 ≠ 3.24

So,
`√((a +b))  \\eq √(a)  + √(b)`