Thread: Surd

How do I simplify to ?

Thanks

Hello, slevvio!

I don't think there is an easy way . . .

First, we must suspect that is a square.

Then we let: .
. . where both are rational numbers.


Square both sides: .

. . and we have:. . a^2 + 2b^2) + 2ab\sqrt{2}" alt="9 + 4\sqrt{2} \;=\a^2 + 2b^2) + 2ab\sqrt{2}" />


These two numbers are equal if their corresponding coefficients are equal.

So we have: .

The second equation gives us: .

Substitute into the first equation: .

Multiply by

. . which factors: .

. . and has roots: .

Since must be rational and the original square root is positive,
. . the only acceptable root is: .


Therefore: .

Originally Posted by Soroban

I don't think there is an easy way . . .

Of course there is one more dear Soroban

Originally Posted by slevvio

How do I simplify to ?

Since the quantity inside bars is positive, we have that

Hello, Krizalid!


Of course, you are right . . . That is a valid method.

I use that method when I'm quite certain that we have a square.


I've done it enough times that I suspect that is a square
. . because: .

And, sure enough: .