Let's simplify the given expression step by step:

The expression is: $\sqrt{17 - \sqrt{77} - \sqrt{78 - 2\sqrt{77}}}$

Step 1: Simplify the inner square root expression

First, let's focus on simplifying the inner square root: $\sqrt{78 - 2\sqrt{77}}$ This expression is in the form $\sqrt{a - 2\sqrt{b}}$, which can sometimes be simplified to the form $\sqrt{(\sqrt{m} - \sqrt{n})^2}$, where $m$ and $n$ are numbers.

Let's try to express $78 - 2\sqrt{77}$ in this form:

Suppose: $\sqrt{m} = \sqrt{x} + \sqrt{y}$ $(\sqrt{x} - \sqrt{y})^2 = x + y - 2\sqrt{xy}$

So, we want: $x + y = 78 \quad \text{and} \quad 2\sqrt{xy} = 2\sqrt{77}$ This implies: $xy = 77$

We need to solve these simultaneous equations:

1. $x + y = 78$
2. $xy = 77$

These correspond to the quadratic equation: $t^2 - 78t + 77 = 0$

Solving this quadratic equation gives us the roots $t = 77$ and $t = 1$, meaning $x = 77$ and $y = 1$ (or vice versa).

Thus: $\sqrt{78 - 2\sqrt{77}} = \sqrt{(\sqrt{77} - \sqrt{1})^2} = \sqrt{77} - 1$

Step 2: Substitute back into the original expression

Now substitute back: $\sqrt{17 - \sqrt{77} - (\sqrt{77} - 1)}$ This simplifies to: $\sqrt{17 - \sqrt{77} - \sqrt{77} + 1} = \sqrt{18 - 2\sqrt{77}}$

Step 3: Simplify $\sqrt{18 - 2\sqrt{77}}$

Similarly, let's try to express $18 - 2\sqrt{77}$ in the form $(\sqrt{x} - \sqrt{y})^2$.

Suppose: $\sqrt{18 - 2\sqrt{77}} = \sqrt{(\sqrt{a} - \sqrt{b})^2}$ This leads to: $a + b = 18 \quad \text{and} \quad 2\sqrt{ab} = 2\sqrt{77}$ Which implies: $ab = 77$

Again solving the quadratic equation: $t^2 - 18t + 77 = 0$

This gives roots $t = 7$ and $t = 11$.

Thus: $\sqrt{18 - 2\sqrt{77}} = \sqrt{(\sqrt{11} - \sqrt{7})^2} = \sqrt{11} - \sqrt{7}$

Final Answer:

$\sqrt{17 - \sqrt{77} - \sqrt{78 - 2\sqrt{77}}} = \sqrt{11} - \sqrt{7}$

Would you like to dive into any part of this process further? Here are some follow-up questions:

1. How does the quadratic formula apply to solving the simplification?
2. Can you explain the process of expressing a difference as a perfect square?
3. What other forms of roots or radical expressions can be simplified similarly?
4. How would the method change if there were additional terms inside the square root?
5. Can you apply this method to another similar expression?

Tip: Always look out for patterns when dealing with nested square roots, as they often simplify into a more manageable form.