How do you simplify sqrt(1+9x^4)?

1 Answer
Aug 8, 2015

That's about as simple is it can be.

You can factor the radicand as

(3x^2-sqrt(6)x+1)(3x^2+sqrt(6)x+1)

but there are no square factors to allow simplification.

Explanation:

If the radicand of a square root has a square factor, then you can move that outside the square root.

For example, sqrt(12) = sqrt(2^2*3) = 2sqrt(3)

With variables you have to be a little more careful, but you can say:

sqrt(9x^2) = 3abs(x)

In the case of (1+9x^4) there is no square factor - the 9 and x^4 do not really help as they are not factors of the whole expression (1+9x^4)

We can factor the radicand to get:

sqrt(1+9x^4) = sqrt((3x^2-sqrt(6)x+1)(3x^2+sqrt(6)x+1))

but I would not call that simpler.