It seems like the expression you're dealing with involves nested square roots. Let's rewrite and simplify it:

\[
6 \times \sqrt{20 - \sqrt{80}}
\]

First, we need to simplify the expression inside the square root:

1. **Simplify \( \sqrt{80} \):**

   \[
   \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}
   \]

   So, the expression becomes:

   \[
   \sqrt{20 - 4\sqrt{5}}
   \]

2. Now, consider simplifying \( \sqrt{20 - 4\sqrt{5}} \). We hypothesize that it can be written in the form \( \sqrt{a} - \sqrt{b} \):

   Let's assume:

   \[
   \sqrt{20 - 4\sqrt{5}} = \sqrt{a} - \sqrt{b}
   \]

   Squaring both sides:

   \[
   20 - 4\sqrt{5} = a + b - 2\sqrt{ab}
   \]

   Now, equate the rational and irrational parts:

   - For the rational part: \( a + b = 20 \)
   - For the irrational part: \( -2\sqrt{ab} = -4\sqrt{5} \)

   Solving the irrational part:

   \[
   2\sqrt{ab} = 4\sqrt{5} \quad \Rightarrow \quad \sqrt{ab} = 2\sqrt{5} \quad \Rightarrow \quad ab = 20
   \]

   Now, solve the system:

   \[
   a + b = 20, \quad ab = 20
   \]

   The solutions for \( a \) and \( b \) are \( a = 10 \) and \( b = 2 \) (or vice versa). So,

   \[
   \sqrt{20 - 4\sqrt{5}} = \sqrt{10} - \sqrt{2}
   \]

3. Now substitute this back into the original expression:

   \[
   6 \times \left(\sqrt{10} - \sqrt{2}\right)
   \]

   So, the simplified form of the expression is:

   \[
   \boxed{6\sqrt{10} - 6\sqrt{2}}
   \]

Would you like more details or have any other questions?

Here are 5 related questions:
1. Simplify \( \sqrt{45 + \sqrt{180}} \).
2. How would you simplify \( \sqrt{12 - \sqrt{32}} \)?
3. What is the simplified form of \( \sqrt{50 + 10\sqrt{5}} \)?
4. How do you simplify \( \sqrt{75 - \sqrt{300}} \)?
5. What is the result of \( 2 \times \sqrt{20 + \sqrt{80}} \)?

**Tip:** When simplifying nested radicals, look for patterns or forms like \( \sqrt{a} \pm \sqrt{b} \) and equate both the rational and irrational parts after squaring.

Explore how to simplify the expression: 6 times square root of 20 minus square root of 80. Learn the step-by-step process to simplify nested radicals, including the use of square root properties and algebraic manipulations. Suitable for advanced high school students.

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