The problem asks to simplify the nested radicals (double square roots). Specifically, it asks to find values to fill in the blanks. (1) $\sqrt{9 + 2\sqrt{20}} = \sqrt{\boxed{22}-1} + \sqrt{\boxed{22}-2}$ (2) $\sqrt{7 - \sqrt{48}} = \sqrt{\boxed{23}-1} - \sqrt{\boxed{23}-2}$

1. Problem Description

The problem asks to simplify the nested radicals (double square roots). Specifically, it asks to find values to fill in the blanks.

(1) $\sqrt{9 + 2\sqrt{20}} = \sqrt{\boxed{22}-1} + \sqrt{\boxed{22}-2}$

(2) $\sqrt{7 - \sqrt{48}} = \sqrt{\boxed{23}-1} - \sqrt{\boxed{23}-2}$

2. Solution Steps

(1)

We want to simplify $\sqrt{9 + 2\sqrt{20}}$ in the form of $\sqrt{a} + \sqrt{b}$.

We can rewrite $9+2\sqrt{20}$ as $(\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}$.

Therefore, we have $a+b = 9$ and $ab = 20$. By observation or solving this system of equations, we find that $a=5$ and $b=4$.

So, $\sqrt{9 + 2\sqrt{20}} = \sqrt{5} + \sqrt{4} = \sqrt{5} + 2$.

We are given that $\sqrt{9 + 2\sqrt{20}} = \sqrt{\boxed{22}-1} + \sqrt{\boxed{22}-2}$.

We want to find a value such that $\sqrt{\boxed{22}-1} + \sqrt{\boxed{22}-2} = \sqrt{5} + 2 = \sqrt{5} + \sqrt{4}$.

Comparing the two expressions, we can see that if we let the value in the box be 6, then $\sqrt{6-1} + \sqrt{6-2} = \sqrt{5} + \sqrt{4} = \sqrt{5} + 2$.

Thus, the value of the box is

6.

(2)

We want to simplify $\sqrt{7 - \sqrt{48}}$. Notice that $\sqrt{48} = \sqrt{4 \cdot 12} = 2\sqrt{12}$.

So we have $\sqrt{7 - 2\sqrt{12}}$.

We want to simplify this in the form $\sqrt{a} - \sqrt{b}$.

We can rewrite $7 - 2\sqrt{12}$ as $(\sqrt{a} - \sqrt{b})^2 = a + b - 2\sqrt{ab}$.

Therefore, we have $a+b = 7$ and $ab = 12$. By observation or solving this system of equations, we find that $a=4$ and $b=3$. Since $a > b$ as $\sqrt{a} - \sqrt{b} > 0$, we can set $a=4$ and $b=3$.

So $\sqrt{7 - \sqrt{48}} = \sqrt{4} - \sqrt{3} = 2 - \sqrt{3}$.

We are given that $\sqrt{7 - \sqrt{48}} = \sqrt{\boxed{23}-1} - \sqrt{\boxed{23}-2}$.

We want to find a value such that $\sqrt{\boxed{23}-1} - \sqrt{\boxed{23}-2} = 2 - \sqrt{3} = \sqrt{4} - \sqrt{3}$.

Comparing the two expressions, we can see that if we let the value in the box be 4, then $\sqrt{4-1} - \sqrt{4-2} = \sqrt{4} - \sqrt{3} = 2 - \sqrt{3}$.

Thus, the value of the box is

4.

3. Final Answer

(1) The value in box 22 is

6. (2) The value in box 23 is

4.