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## Main Question or Discussion Point

I want to de-nest the following radical:

(1) [tex] \sqrt{3+2\sqrt{2}} [/tex]

Into the general simplified form:

(2) [tex] a+b\sqrt{2} [/tex]

Equating (1) with (2),

(3) [tex] \sqrt{3+2\sqrt{2}} = a+b\sqrt{2} [/tex]

and squaring both sides:

(4) [tex] 3+2\sqrt{2} = a^2 + 2b^2 + 2ab\sqrt{2} [/tex]

generates a system of two equations with two unknowns, a and b, after equating the rational and irrational parts:

(5) [tex] 3 = a^2 + 2b^2 [/tex]

(6) [tex] 2\sqrt{2} = 2ab\sqrt{2} [/tex]

Simplifying (6) and solving for b:

(7) [tex] b=\frac{1}{a} [/tex]

Substituting (7) into (5) yields:

(8) [tex] 3 = a^2 + 2\frac{1}{a^2} [/tex]

Clearing the denominator and moving non-zero terms to one side generates a quartic equation:

(9) [tex] 0 = a^4 - 3a^2 +2 [/tex]

that can be made into a quadratic equation with the substitution:

(10) [tex] x=a^2, x^2 = (a^2)^2 = a^4, x = \pm \sqrt{a} [/tex]

(11) [tex] 0 = x^2 - 3x + 2 [/tex]

The square root of the discriminant is an integer, 1, which presumably makes simplification of the nested radical possible. Finding the roots, x, of (11) gives

(12)[tex] x=1, x =\sqrt{2} [/tex]

Substituting the roots in (12) into (10) gives a:

(13) [tex] a = \pm \sqrt{2} , a = \pm 1 [/tex]

Then b is found from (7):

(14) [tex] b = \pm \frac{1}{\sqrt{2}} , b = \pm 1 [/tex]

Using the positive values of a and b, the de-nested radical is:

(15) [tex] \sqrt{3+2\sqrt{2}} = 1+\sqrt{2} [/tex]

My questions are:

(1) Is this approach for simplifying nested radical correct?

(2) The positive values of a and b produce the correct simplified form in (15) while the negative values of a and b do not. Is there a way to figure out which roots from (9) are correct and which to reject other than calculating the numerical value of (15) with both positive and negative a's and b's to see which is equal to the nested form?

(3) I have also seen the general simplified de-nested form as:

(16) [tex] \sqrt{a} + \sqrt{b} [/tex]

Going through an analogous process as above, it generates the correct simplified form in (15) as well. Is one form (2) or (15) better than the other? Or is it just a personal preference which one to use?

Thanks!

(1) [tex] \sqrt{3+2\sqrt{2}} [/tex]

Into the general simplified form:

(2) [tex] a+b\sqrt{2} [/tex]

Equating (1) with (2),

(3) [tex] \sqrt{3+2\sqrt{2}} = a+b\sqrt{2} [/tex]

and squaring both sides:

(4) [tex] 3+2\sqrt{2} = a^2 + 2b^2 + 2ab\sqrt{2} [/tex]

generates a system of two equations with two unknowns, a and b, after equating the rational and irrational parts:

(5) [tex] 3 = a^2 + 2b^2 [/tex]

(6) [tex] 2\sqrt{2} = 2ab\sqrt{2} [/tex]

Simplifying (6) and solving for b:

(7) [tex] b=\frac{1}{a} [/tex]

Substituting (7) into (5) yields:

(8) [tex] 3 = a^2 + 2\frac{1}{a^2} [/tex]

Clearing the denominator and moving non-zero terms to one side generates a quartic equation:

(9) [tex] 0 = a^4 - 3a^2 +2 [/tex]

that can be made into a quadratic equation with the substitution:

(10) [tex] x=a^2, x^2 = (a^2)^2 = a^4, x = \pm \sqrt{a} [/tex]

(11) [tex] 0 = x^2 - 3x + 2 [/tex]

The square root of the discriminant is an integer, 1, which presumably makes simplification of the nested radical possible. Finding the roots, x, of (11) gives

(12)[tex] x=1, x =\sqrt{2} [/tex]

Substituting the roots in (12) into (10) gives a:

(13) [tex] a = \pm \sqrt{2} , a = \pm 1 [/tex]

Then b is found from (7):

(14) [tex] b = \pm \frac{1}{\sqrt{2}} , b = \pm 1 [/tex]

Using the positive values of a and b, the de-nested radical is:

(15) [tex] \sqrt{3+2\sqrt{2}} = 1+\sqrt{2} [/tex]

My questions are:

(1) Is this approach for simplifying nested radical correct?

(2) The positive values of a and b produce the correct simplified form in (15) while the negative values of a and b do not. Is there a way to figure out which roots from (9) are correct and which to reject other than calculating the numerical value of (15) with both positive and negative a's and b's to see which is equal to the nested form?

(3) I have also seen the general simplified de-nested form as:

(16) [tex] \sqrt{a} + \sqrt{b} [/tex]

Going through an analogous process as above, it generates the correct simplified form in (15) as well. Is one form (2) or (15) better than the other? Or is it just a personal preference which one to use?

Thanks!