# Help with a surds problem?

• Aug 19th 2012, 04:58 AM
ACM
Help with a surds problem?

If √m +√n = √(7+√48) calculate to the nearest digit m2 + n2 ?

Thanks
(Worried)
• Aug 19th 2012, 08:10 AM
ACM
Re: Help with a surds problem?

If m and n are rational numbers, √m +√n = √(7+√48) calculate to the nearest digit m2 + n2 ?

Thanks
• Aug 19th 2012, 04:05 PM
Soroban
Re: Help with a surds problem?
Hello, ACM!

Quote:

$\displaystyle \text{If }\sqrt{m} + \sqrt{n} \:=\:\sqrt{7+\sqrt{48}},\,\text{calculate to the nearest digit }m^2+n^2.$

Whenever I see that an expression under a square root contains a square root,
. . I always wonder if that expression is a square.

Inside, we have: .$\displaystyle 7 + \sqrt{48} \:=\:7 + 4\sqrt{3}$ . . . . which happens to be $\displaystyle (2 + \sqrt{3})^2$

The problem becomes: .$\displaystyle \sqrt{m}+\sqrt{n} \;=\;\sqrt{(2 + \sqrt{3})^2}$

. . . . . . . . . . . . . . . . . .$\displaystyle \sqrt{m}+\sqrt{n} \;=\;2 + \sqrt{3}$

Hence: .$\displaystyle \begin{Bmatrix}m &=& 4 \\ n &=& 3\end{Bmatrix}$ . or vice versa.

Therefore: .$\displaystyle m^2 + n^2 \;=\;4^2 + 3^2 \;=\;25$