2026/070) Simplify $\sqrt{8+\sqrt{7}} - \sqrt{8-\sqrt{7}}$

Because $8 + 2 \sqrt(7)$ has $\sqrt(7)$ as one of the terms so the square root is of the form $a + b \sqrt(7)$ where a and b are rational numbers

Squaring we get

$a^2 + 7b^2 + 2ab \sqrt(7) = 8 + 2 \sqrt(7)$

So comparing rational and irrational parts we get

$a^2 + 7b^2 = 8\cdots(1)$

and $ab = 1\cdots(2)$

So a and b both are positive (-ve shall give -ve square root)

From (2) we get

$b = \frac{1}{a}$

Putting in (1) we get

$a^2 + 7 \frac{1}{a^2} = 8$

or $a^4 - 8a^2 +7=0$

or $(a^2-1)(a^2-7) = 0$

as a is rational so $a^2-7=0$ is ruled out and we have $a^2-1=0$

As a is positive a = 1 and so b = 1 from (2)

So $\sqrt{8+\sqrt{7}} = 1+ \sqrt{7}\cdots(3)$

Now Similarly $\sqrt{8 -\sqrt{7}} = \pm (1- \sqrt{7})$ we need to chooses the proper sign  

We need to take the principal root that is the value

 $\sqrt{8 -\sqrt{7}} = \sqrt{7}-1\cdots(4)$

from (3) and (4)

 $\sqrt{8+\sqrt{7}} - \sqrt{8-\sqrt{7}}= 2$