Are the two numbers
square root of 3 + and square root of 11 and
square root of 5 + square root of 8 equal? If not, which is larger?
Hello, matgrl!
$\displaystyle \text{Are the two numbers }\sqrt{3} + \sqrt{11}\,\text{ and }\,\sqrt{5} + \sqrt{8}\,\text{ equal?}$
$\displaystyle \text{If not, which is larger?}$
We have: . . . .$\displaystyle \sqrt{3} + \sqrt{11} \quad[?]\quad \sqrt{5} + 2\sqrt{2}$
Square: . .$\displaystyle 3 + 2\sqrt{33} + 11 \quad [?]\quad 5 + 4\sqrt{10} + 8 $
. . . . . . . . . . . $\displaystyle 1 + 2\sqrt{33} \quad [?] \quad 4\sqrt{10}$
Square: .$\displaystyle 1 + 4\sqrt{33} + 132 \quad [?]\quad 160$
. . . . . . . . . . . . . .$\displaystyle 4\sqrt{33} \quad[?]\quad 27 $
Square: . . . . . . . . . $\displaystyle 528 \quad[?]\quad 729$
Since 528 < 729, the original statement is: .$\displaystyle \sqrt{3} + \sqrt{11} \;<\;\sqrt{5} + \sqrt{8}$