We are tasked with simplifying the following expression step by step:

$\frac{4 + 4\sqrt{6}}{1 - \sqrt{6}}$

Step 1: Recognize the need to rationalize the denominator.

The denominator $1 - \sqrt{6}$ includes a square root, so we rationalize it by multiplying both the numerator and denominator by the conjugate of $1 - \sqrt{6}$, which is $1 + \sqrt{6}$.

$\text{Multiply both numerator and denominator by } (1 + \sqrt{6}):$ $\frac{4 + 4\sqrt{6}}{1 - \sqrt{6}} \cdot \frac{1 + \sqrt{6}}{1 + \sqrt{6}}$

Step 2: Expand both numerator and denominator.

Numerator:

Using the distributive property: $(4 + 4\sqrt{6})(1 + \sqrt{6}) = 4(1) + 4(\sqrt{6}) + 4\sqrt{6}(1) + 4\sqrt{6}(\sqrt{6})$ Simplify: $4 + 4\sqrt{6} + 4\sqrt{6} + 4 \cdot 6 = 4 + 8\sqrt{6} + 24$ Combine like terms: $4 + 24 + 8\sqrt{6} = 28 + 8\sqrt{6}$

Denominator:

Using the difference of squares formula: $(1 - \sqrt{6})(1 + \sqrt{6}) = 1^2 - (\sqrt{6})^2$ Simplify: $1 - 6 = -5$

Step 3: Rewrite the expression.

After expanding, the expression becomes: $\frac{28 + 8\sqrt{6}}{-5}$

Step 4: Simplify by distributing the denominator.

Divide each term in the numerator by $-5$: $\frac{28}{-5} + \frac{8\sqrt{6}}{-5}$

Simplify further: $-\frac{28}{5} - \frac{8\sqrt{6}}{5}$

Final Answer:

$\boxed{-\frac{28}{5} - \frac{8\sqrt{6}}{5}}$

If you'd like, I can verify or explore further aspects of this problem.

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Follow-Up Questions:

1. What is the purpose of rationalizing the denominator in expressions with square roots?
2. Can you explain the difference of squares used to simplify the denominator?
3. Why does distributing the denominator ensure the expression remains equivalent?
4. How does conjugation help in simplifying square root terms?
5. What happens if the denominator was $1 + \sqrt{6}$ instead?

Tip:

When dealing with expressions that involve square roots, rationalizing the denominator makes them easier to interpret and compare.