Let's find the exact value of [tex]\(\sqrt{2+\sqrt{3}} - \sqrt{2-\sqrt{3}}\)[/tex].
First, define [tex]\(x\)[/tex] as:
[tex]\[ x = \sqrt{2+\sqrt{3}} - \sqrt{2-\sqrt{3}} \][/tex]
Next, square both sides of the equation to eliminate the square roots:
[tex]\[ x^2 = (\sqrt{2+\sqrt{3}} - \sqrt{2-\sqrt{3}})^2 \][/tex]
Using the algebraic identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[ x^2 = (\sqrt{2+\sqrt{3}})^2 - 2(\sqrt{2+\sqrt{3}})(\sqrt{2-\sqrt{3}}) + (\sqrt{2-\sqrt{3}})^2 \][/tex]
Calculate the squares of each term:
[tex]\[ x^2 = (2+\sqrt{3}) + (2-\sqrt{3}) - 2(\sqrt{(2+\sqrt{3})(2-\sqrt{3})}) \][/tex]
Simplify the expression by combining like terms:
[tex]\[ x^2 = 2 + \sqrt{3} + 2 - \sqrt{3} - 2\sqrt{(2+\sqrt{3})(2-\sqrt{3})} \][/tex]
Notice that [tex]\(\sqrt{3}\)[/tex] and [tex]\(-\sqrt{3}\)[/tex] cancel each other out:
[tex]\[ x^2 = 4 - 2\sqrt{(2+\sqrt{3})(2-\sqrt{3})} \][/tex]
Simplify the term inside the square root:
[tex]\[ x^2 = 4 - 2\sqrt{4 - (\sqrt{3})^2} \][/tex]
Since [tex]\((\sqrt{3})^2 = 3\)[/tex]:
[tex]\[ x^2 = 4 - 2\sqrt{4 - 3} \][/tex]
[tex]\[ x^2 = 4 - 2\sqrt{1} \][/tex]
[tex]\[ x^2 = 4 - 2 \][/tex]
Finally, simplify the expression:
[tex]\[ x^2 = 2 \][/tex]
Taking the positive square root of both sides (since the expression inside the square roots is positive):
[tex]\[ x = \sqrt{2} \][/tex]
Therefore, the exact value of [tex]\(\sqrt{2+\sqrt{3}} - \sqrt{2-\sqrt{3}}\)[/tex] is:
[tex]\[ \sqrt{2} \][/tex]
Additionally, the numerical value of [tex]\(\sqrt{2}\)[/tex] is approximately [tex]\(1.4142135623730951\)[/tex].
Thus, the desired value of [tex]\(\sqrt{2+\sqrt{3}} - \sqrt{2-\sqrt{3}}\)[/tex] is:
[tex]\[ \boxed{\sqrt{2}} \][/tex]
And numerically:
[tex]\[ \boxed{1.4142135623730951} \][/tex]
