Question
Find the value of the square root \sqrt{3+2\sqrt{2}}
Find the value of the square root \sqrt{3+2\sqrt{2}}
Let
3+2\sqrt{2} = (a+b\sqrt{2})^2
3-2\sqrt{2} = (a-b\sqrt{2})^2
in which a and b are positive integers
Addition of the two equations gives
(a+b\sqrt{2})^2+(a-b\sqrt{2})^2 = 6
Expand and simplify the equation
2a^2+2\cdot 2b^2=6
Multiplying the two equations gives
(3+2\sqrt{2})(3-2\sqrt{2}) = (a+b\sqrt{2})^2 (a-b\sqrt{2})^2
Simplify the equation using difference of square formula
(a^2 - b^2\cdot 2)^2 = 3^2 - 2^2\cdot 2=1
Taking the square root
2 systems of equations are obtained from (1) and (2)
Case 1
\begin{cases} a^2+2b^2&=3 \\ a^2 - 2b^2 &=1 \end{cases}
Solving the system of equaitons gives
a = \sqrt{\dfrac{3+1}{2}} =\sqrt{2}
b = \sqrt{\dfrac{3-1}{2\cdot 2}}=\sqrt{\dfrac{1}{2}}
Case 2
\begin{cases} a^2+2b^2&=3\\ a^2 - 2b^2 &= -1 \end{cases}
Solving the system of equaitons gives
a = 1, b=1
Both results give the same root. Therefore,
The value of the square root of
\sqrt{3+2\sqrt{2}}=\sqrt{(1+\sqrt{2})^2}=1+\sqrt{2}
\sqrt{3-2\sqrt{2}}=\sqrt{(1-\sqrt{2})^2}=\sqrt{2}-1