Answer:
\(1+\sqrt{2}\)
Step-by-step explanation:
We have been given a radical expression \(\sqrt{3+2\sqrt{2}}\). We are asked to simplify the given expression.
We will add \((\sqrt{2})^2-2\) to our given expression as after adding and subtracting same quantity the value of our expression will be same.
\(\sqrt{3+2\sqrt{2}+(\sqrt{2})^2-2}\)
\(\sqrt{3-2+2\sqrt{2}+(\sqrt{2})^2}\)
\(\sqrt{1+2\sqrt{2}+(\sqrt{2})^2}\)
Using perfect square formula \((a+b)^2=a^2+2ab+b^2\) we can rewrite our expression as:
\(\sqrt{1^2+2\sqrt{2}\cdot 1+(\sqrt{2})^2}\)
\(\sqrt{(1+\sqrt{2})^2}\)
Applying radical rule \(\sqrt[n]{x^n} =x\), we will get,
\((1+\sqrt{2})^{\frac{2}{2}}=1+\sqrt{2}\)
Therefore, the simplified form of our given expression would be \(1+\sqrt{2}\).
