Let's simplify the expression $2\sqrt{5\left(3\sqrt{10-4\sqrt{5)}}\right)}$ step by step.

Step 1: Simplify the expression inside the square root

Start by simplifying the innermost expression: $\sqrt{10 - 4\sqrt{5}}$

We can try expressing this as: $\sqrt{a} - \sqrt{b}$ where $a$ and $b$ are positive numbers.

Square both sides: $10 - 4\sqrt{5} = a + b - 2\sqrt{ab}$

Comparing both sides, we get two equations:

1. $a + b = 10$
2. $-2\sqrt{ab} = -4\sqrt{5}$

From the second equation: $\sqrt{ab} = 2\sqrt{5}$ Square both sides: $ab = 20$

Now, solve the system:

1. $a + b = 10$
2. $ab = 20$

These are the roots of the quadratic equation: $t^2 - 10t + 20 = 0$

Solving this quadratic equation: $t = \frac{10 \pm \sqrt{100 - 80}}{2} = \frac{10 \pm \sqrt{20}}{2} = \frac{10 \pm 2\sqrt{5}}{2}$ $t = 5 \pm \sqrt{5}$

So, $a = 5 + \sqrt{5}$ and $b = 5 - \sqrt{5}$.

Thus: $\sqrt{10 - 4\sqrt{5}} = \sqrt{5+\sqrt{5}} - \sqrt{5-\sqrt{5}}$

Step 2: Substitute and simplify the expression

Substitute this into the original expression: $2\sqrt{5\left(3\sqrt{5+\sqrt{5}} - \sqrt{5-\sqrt{5}}\right)}$

This expression can get quite complicated without specific substitutions or identities, but typically the strategy would involve:

• Trying to simplify each radical,
• Looking for patterns or symmetry,
• Applying approximations if needed.

However, it's possible that further simplifications might only be numerical unless a specific identity can be applied.

Do you want to delve further into each step or explore approximations?

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Related Questions:

1. How would you solve $\sqrt{10 - 4\sqrt{5}}$ by completing the square?
2. What techniques can simplify nested radicals effectively?
3. How do you solve the quadratic equation $t^2 - 10t + 20 = 0$?
4. How does the quadratic formula work, and how does it relate to this problem?
5. What are some common strategies for dealing with expressions under square roots in general?

Tip:

For nested radicals, it's often helpful to first test if they can be expressed as a difference or sum of two square roots. This can make the overall simplification process much easier.