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The expression to simplify is:

\[
D = \sqrt{14 + 2\sqrt{76 - 4\sqrt{7} - \sqrt{6 - 2\sqrt{7}}}}
\]

Let's break it down step by step.

### Step 1: Simplify the innermost square root
Start with the innermost expression \(\sqrt{6 - 2\sqrt{7}}\).

We can assume that it can be expressed as:

\[
\sqrt{a} - \sqrt{b}
\]

Squaring both sides:

\[
(\sqrt{a} - \sqrt{b})^2 = a + b - 2\sqrt{ab}
\]

This should equal to \(6 - 2\sqrt{7}\):

\[
a + b = 6 \quad \text{and} \quad -2\sqrt{ab} = -2\sqrt{7}
\]

From \(-2\sqrt{ab} = -2\sqrt{7}\), we get \(ab = 7\).

Now we solve \(a\) and \(b\) from:

\[
a + b = 6, \quad ab = 7
\]

The roots of the quadratic equation \(x^2 - 6x + 7 = 0\) are:

\[
x = \frac{6 \pm \sqrt{6^2 - 4 \times 1 \times 7}}{2} = \frac{6 \pm \sqrt{36 - 28}}{2} = \frac{6 \pm \sqrt{8}}{2} = \frac{6 \pm 2\sqrt{2}}{2} = 3 \pm \sqrt{2}
\]

So, \(a = 3 + \sqrt{2}\) and \(b = 3 - \sqrt{2}\).

Thus:

\[
\sqrt{6 - 2\sqrt{7}} = \sqrt{3 + \sqrt{2}} - \sqrt{3 - \sqrt{2}}
\]

### Step 2: Substitute and simplify the expression
Next, substitute this into the original expression for \(D\):

\[
D = \sqrt{14 + 2\sqrt{76 - 4\sqrt{7} - (\sqrt{3 + \sqrt{2}} - \sqrt{3 - \sqrt{2}})}}
\]

This expression still needs more work to fully simplify, as it involves nested radicals.

Let me know if you'd like to proceed with further simplifications or if you need detailed steps for any part of this process!

Learn how to simplify nested radicals in mathematical expressions with a step-by-step solution. This problem involves square roots and nested radicals, demonstrating advanced algebraic techniques. Suitable for advanced high school students.

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