Problem with $\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}$

Question

How to simplify $$\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}?$$

Rationalise the denominator

$$\frac{\sqrt{6+4\sqrt{2}}}{4}(2-\sqrt{2})$$

This is still not simplify.

Answer 1

Hint. Note that $$(2\pm\sqrt{2})=\sqrt{(2\pm\sqrt{2})^2}=\sqrt{6\pm 4\sqrt{2}}.$$

Answer 2

Hint: $6+4\sqrt{2} = (2+\sqrt{2})^2$.

Answer 3

$\sqrt{6+4\sqrt{2}} = \sqrt{(2 + \sqrt{2})^{2}} = {2+\sqrt{2}}$ ,

as $6+4\sqrt{2}= 4 +2 +2.2\sqrt{2} = (\sqrt{2})^{2} + 2 .2. \sqrt{2} + 2^ 2 = (2 + \sqrt{2})^{2}$

Now $\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}} = \frac{2+\sqrt{2}}{2(2+\sqrt{2})}= \frac{1}{2}$

Answer 4

We have $$6+4\sqrt{2}=4+2+4\sqrt{2}$$ so $$2\sqrt{6+4\sqrt{2}}=4+2\sqrt{2}$$ and we get

$$\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}=\frac{1}{2}$$