Let's address the problem step by step:
### Part a: Finding the positive square root of [tex]\( 3 + 2\sqrt{2} \)[/tex]
To find the square root, we denote:
[tex]\[ \sqrt{3 + 2\sqrt{2}} = x \][/tex]
Squaring both sides, we obtain:
[tex]\[ x^2 = 3 + 2\sqrt{2} \][/tex]
We need to find the value of [tex]\( x \)[/tex] that is positive and satisfies this equation.
The positive square root of [tex]\( 3 + 2\sqrt{2} \)[/tex] can be directly given as:
[tex]\[ x \approx 2.414213562373095 \][/tex]
So, the positive square root of [tex]\( 3 + 2\sqrt{2} \)[/tex] is approximately [tex]\( 2.414213562373095 \)[/tex].
### Part b: Finding the positive square root of [tex]\( 3 - 2\sqrt{6} \)[/tex]
Similarly, we denote:
[tex]\[ \sqrt{3 - 2\sqrt{6}} = y \][/tex]
Squaring both sides, we obtain:
[tex]\[ y^2 = 3 - 2\sqrt{6} \][/tex]
Now we need to find the value of [tex]\( y \)[/tex] that satisfies this equation.
Since [tex]\( 3 - 2\sqrt{6} \)[/tex] is a negative number which is approximately [tex]\(-4.898979485566356\)[/tex], its square root would involve imaginary numbers. The positive square root being considered in terms of complex numbers can be expressed as [tex]\( y \approx (8.438028591668086 \times 10^{-17} + 1.3780346459963755i) \)[/tex].
Here, the root is complex because the expression [tex]\( 3 - 2\sqrt{6} \)[/tex] is negative inside the square root.
Thus, the positive square root of [tex]\( 3 - 2\sqrt{6} \)[/tex] is approximately [tex]\( (8.438028591668086 \times 10^{-17} + 1.3780346459963755i) \)[/tex].
