Answer :
Recall that
[tex]\|\mathbf u\times\mathbf v\|=\|\mathbf u\|\|\mathbf v\|\sin\theta[/tex]
where [tex]\theta[/tex] is the angle between the vectors [tex]\mathbf u[/tex] and [tex]\mathbf v[/tex]. No need to actually compute the cross product.
We can find the angle between the vectors using the dot product formula,
[tex]\mathbf u\cdot\mathbf v=\|\mathbf u\|\|\mathbf v\|\cos\theta[/tex]
[tex]\implies\theta=\cos^{-1}\left(\dfrac{(2\mathbf i+2\mathbf j-\mathbf k)\cdot(-\mathbf i+\mathbf k)}{\sqrt{2^2+2^2+(-1)^2}\sqrt{(-1)^2+1^2}}\right)=\cos^{-1}\left(-\dfrac1{3\sqrt2}\right)[/tex]
Then
[tex]\|\mathbf u\times\mathbf v\|=3\sqrt2\sin\theta=\boxed{\sqrt{17}}[/tex]