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

Aug 8, 2015

That's about as simple is it can be.

You can factor the radicand as

$\left(3 {x}^{2} - \sqrt{6} x + 1\right) \left(3 {x}^{2} + \sqrt{6} x + 1\right)$

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, $\sqrt{12} = \sqrt{{2}^{2} \cdot 3} = 2 \sqrt{3}$

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

$\sqrt{9 {x}^{2}} = 3 \left\mid x \right\mid$

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

We can factor the radicand to get:

$\sqrt{1 + 9 {x}^{4}} = \sqrt{\left(3 {x}^{2} - \sqrt{6} x + 1\right) \left(3 {x}^{2} + \sqrt{6} x + 1\right)}$

but I would not call that simpler.